Praise be to Allah.
It is not permissible for a man who is in ihram to wear socks, but it is permissible for a woman to do that .
Shaykh Ibn ‘Uthaymin said in al-Sharh al-Mumti’ (7/154): Question: is it haram for her (i.e., the woman in ihram) to wear socks?
“The answer is no. Socks are haram for men (in ihram).
If a man in ihram needs to wear socks , it is permissible for him to wear them, but he has to offer the fidyah (ransom), which is either to sacrifice a sheep, or to feed six poor persons, or to fast for three days; he may do whichever he wants of these three.”
The Standing Committee was asked: what is the ruling on wearing socks on the feet and doing tawaf al-qudum in them during Hajj, and doing the tawaf of `Umrah in them during `Umrah?
“It is not permissible for men to wear socks when in ihram for Hajj or `Umrah. If he needs to wear them because of sickness etc, then he must offer a ransom (fidyah) , which is to fast for three days, or to feed six poor persons, giving each one half a sa’ of dates or similar food, or to sacrifice a sheep.” (Fatawa al-Lajnah al-Daimah li’l-Buhuth al-‘Ilmiyyah wa’l-Ifta, 11/183)
And Allah knows best.