Answer
I say, and with God's help: It is not permissible to wipe over thin socks by consensus in our four schools of thought, and they did not rely on the hadith of Al-Mughirah ibn Shu'bah : "Indeed, the Messenger of Allah performed ablution and wiped over the socks and the sandals" in Sahih Ibn Khuzaymah 1: 99, Sahih Ibn Hibban 4: 167, Jami' at-Tirmidhi 1: 167, and he authenticated it, and Sunan Abu Dawood 1: 41, and Sunan an-Nasa'i al-Kubra 1: 92, and Sunan Ibn Majah 1: 185; for several reasons:
1. The Quran commanded us to wash the feet, and it is not permissible for us to abandon a command in the Quran based on a single hadith.
2. Wiping over the leather socks has been established by a mutawatir or well-known hadith, as Al-Suyuti said in "Tadrib al-Rawi" 2: 179, and "Al-Azhar al-Mutanathira" in the mutawatir reports, as it was narrated by seventy companions, and Al-Ayni mentioned in his book "Al-Binayah" 1: 554, and "Sharh Ma'ani al-Athar" about sixty-seven companions, and similar reports are added to the Quran. Abu Hanifah said: "I did not say wiping over the leather socks until I received reports that are clearer than the sun," and he also said: "I fear disbelief for whoever does not see wiping over the leather socks"; because the reports that came regarding it were in the realm of mutawatir: that is, the moral mutawatir, even if they were from the singular verbal reports, as in "Fath Bab al-'Inayah" 1: 183.
3. Wiping over the leather sock is based on preference; because the analogy is to wash the feet, and preference cannot be based on analogy, but rather includes what is considered from its individuals. Thick socks may be wiped over if they meet the conditions for wiping over the leather sock. The leather sock is what covers the ankle, as mentioned in "Sharh al-Wiqayah" p. 114.
Thick socks are those worn like leather socks made of linen, cotton, wool, hair, felt, or other materials that meet the conditions for permissible wiping over them, as mentioned in "Sharh al-Wiqayah" p. 114. This is what can be understood from the words of Ibn Abidin in "Radd al-Muhtar" 1: 179, and Al-Nabulsi in "Nihayat al-Marad" p. 388.
The conditions for wiping over the socks are:
1) The conditions for wiping over the leather socks include wearing them in a state of purity, covering the ankles, the ability to walk regularly in them for more than a distance of a farsakh, their adherence to the feet without tightness, and preventing water from reaching the body, among others.
2) They must be either sandal-like or leathered; because it is possible to regularly walk in them, and the concession is for this reason, thus it becomes like the leather sock, as in "Tafsir al-Haqaiq" 1: 52. The sandal is one that has leather placed on its bottom like a sandal for the foot in the apparent narration, and in the narration of Al-Hasan: it should reach the ankle. See: "Al-Iyadah" Q7/B, and "Tafsir" 1: 52, and "Nihayat al-Marad" p. 389.
The leathered one is one that has leather placed on its top and bottom, as in "Al-Iyadah" Q7/B, and "Tafsir" 1: 52, and "Nihayat al-Marad" p. 389.
3) It must be thick if it is not leathered or sandal-like, as in "Radd al-Muhtar" 1: 179, and the limit of thickness is as follows:
A. That what is underneath them is not visible to the observer.
B. That it is not transparent and does not conceal what is behind it, so that water does not pass through them, as in "Al-Hadiyyah al-'Alaiyyah" p. 39, and "Bada'i al-Sana'i" 1: 10.
C. That it adheres to the leg without tying, as in "Al-Durr al-Mukhtar" 1: 179, and "Nihayat al-Marad" p. 388, and "Tafsir" 1: 52.
Al-Kasani said in "Bada'i al-Sana'i" 1: 10: "As for wiping over the socks, if they are leathered or sandal-like, it suffices without disagreement among our scholars. If they are thin and allow water to pass through, it is not permissible to wipe over them by consensus. If they are thick, it is not permissible according to Abu Hanifah , and according to Abu Yusuf and Muhammad : it is permissible. It was narrated from Abu Hanifah that he retracted his statement regarding them in his later years, as he wiped over his socks during his illness, and then said to his visitors: 'I did what I used to prevent people from doing.' They used this as evidence for his retraction, and according to their view, fatwas are given as in "Sharh al-Wiqayah" p. 115, and "Al-Ikhtiyar" 1: 36. Ismail Al-Nabulsi said: and the most correct is his retraction as in "Al-Majma'" and "Durar al-Bihar", and in "Al-Khulasa": and it is said that he retracted, and upon it is the fatwa, and in "Tafsir" 1: 52: it is narrated that Abu Hanifah retracted to their view seven days before his death, and in "Al-Nawadir": three days, and it is said: seven, and upon it is the fatwa, and similarly in "Al-Dhakirah", and the jurist Abu Lays said: and this is what we take from, as in "Nihayat al-Marad" p. 388.
Al-Nur Al-Din Ittar said in "I'lam al-Anam Sharh Bulugh al-Maram" 1: 187-188: "The Malikis and Shafi'is prohibited wiping over the socks based on the apparent meaning of the verse of ablution, which is also the opinion of Imam Abu Hanifah. They did not act upon the hadith of Al-Mughirah regarding wiping over the socks, and their excuse for that is clear.
However, we see the possibility of acting upon the hadith if we refer it to the original matter, which is wiping over the leather socks. If the sock meets the characteristics of the leather sock, we permit wiping over it; otherwise, it is not permissible, and this is the path of Imam Ahmad and the companions of Abu Hanifah, Abu Yusuf and Muhammad, and upon it is the fatwa in the Hanafi school."
4. No one among the imams has said that wiping over the sandals is permissible, as in "Ma'arif al-Sunan" 1: 347, and they interpreted the meaning of this word as what Al-Tahawi said in "Sharh Ma'ani al-Athar" 1: 97: "It is possible that the Messenger of Allah wiped over sandals under which there were socks, and he intended by that wiping to his socks, not to his sandals, and if he were wearing them without sandals, it would be permissible for him to wipe over them, so his wiping was intended for the socks, and his wiping over the sandals was additional." This indicates that the apparent meaning of the hadith is not acted upon.
5. This hadith has been rejected by the major scholars of hadith:
Abu Dawood said in his Sunan 1: 41: "Abdurrahman ibn Mahdi would not narrate this hadith; because it is known about Al-Mughirah that the Prophet wiped over the leather socks." Al-Bayhaqi said: "It is a rejected hadith, weakened by Sufyan Al-Thawri, Abdurrahman ibn Mahdi, Ahmad ibn Hanbal, Yahya ibn Ma'in, Ali ibn Al-Madini, and Muslim ibn Al-Hajjaj, and it is known about Al-Mughirah that he narrated the hadith of wiping over the leather socks, and it is reported from a group that they did it."
Al-Nawawi said: "Each one of these, if he were alone, would have precedence over Al-Tirmidhi, especially since the defect is prioritized over the approval," and he said: "The scholars agreed on its weakness, and the statement of Al-Tirmidhi: 'It is good and authentic' is not accepted." The complete discussion is in "Nasb al-Rayah" 1: 184, and "Ma'arif al-Sunan" 1: 349, and "Tuhfat al-Ahwadhi" 1: 278.
6. It contradicts the apparent meaning of the Quran regarding the obligation to wash the feet, as Muslim ibn Al-Hajjaj weakened this report, saying: "Abu Qais Al-Awdi and Hudhail ibn Sharhabil cannot be relied upon, especially with their contradiction to the eminent scholars who narrated this report from Al-Mughirah, as they said: He wiped over the leather socks, and we do not abandon the apparent meaning of the Quran for someone like Abu Qais and Hudhail," as in "Nasb al-Rayah" 1: 184, and "Ma'arif al-Sunan" 1: 349, unlike wiping over the leather socks, which the ummah received with acceptance due to the mutawatir narration regarding it, as in "Ma'arif al-Sunan" 1: 350.
Al-Binuri said in "Ma'arif al-Sunan" 1: 350-351: "In summary, they did not act upon the generality of the hadith, but it seems that they acted upon refining the criteria for the leather sock, thus including what we mentioned. In any case, if the hadith of the socks is authentic, it cannot be acted upon in its generality, which includes both thick and thin socks, due to the opposition of the recited Quran. Yes, they acted upon part of it, either by adhering to it or by refining the leather sock mentioned in the mutawatir reports."
7. Wiping over the sock was reported only once from Al-Mughirah , and it was narrated from him in about sixty ways regarding wiping over the leather socks, thus it is an anomaly and an error from the narrator. Al-Binuri said in "Ma'arif al-Sunan" 1: 350-351: "The hadith is narrated from Al-Mughirah in about sixty ways, and the wording of the hadith in question is mentioned only in this way, so how can one feel assured by it? Moreover, the practice of some lenient people in wiping over thin socks has no basis in the Shari'ah to rely upon. If it is based on this hadith, you have known what is in it and what the imams said, and if it is based on the words of the jurists, they stipulated either leathering or sandal-like, or at least thickness."
8. The hadith did not show any characteristic of the leather sock, so we cannot say that it implies wiping over thin socks; therefore, it should be interpreted with other hadiths that state the permissibility of wiping over the leather sock, so the sock must meet the conditions required for the leather sock to permit wiping over it. Al-Nur Al-Din Ittar said in "I'lam al-Anam Sharh Bulugh al-Maram" 1: 187-188: "Some scholars have relied on this hadith and permitted wiping over the socks regardless of their condition, and if you contemplate the hadith, you will find it narrating an actual incident that does not clarify the details of the characteristics of the sock that the Prophet wiped over: What is its thickness? What is its durability? It could be above the leather sock or have a sole, or it might not be so. It is known in the principles of jurisprudence that inferring from actual incidents requires knowledge of their circumstances and contexts.
The truth is that it is not correct to use this hadith as evidence for what they have claimed, and it is a neglect on their part and a leniency regarding the Shari'ah. It has become clear that wiping over the leather socks has been established by mutawatir Sunnah and by consensus on its permissibility... and that it is not correct to use ... as evidence for the permissibility of wiping over the socks in an unrestricted manner without conditions." And Allah knows best.