As far as Islamic sharia is concerned, it is permissible to wipe with wet hands over leather socks, instead of washing the feet when this is done in conformity with the prescribed conditions prescribed by sharia.
However, scholars differ in opinion about the permissibility of wiping on other types of socks (cloth, cotton, and nylon). Reliable scholars and mujtahids like Ibn Taymiyah have affirmed the permissibility of wiping over such socks provided they are thick and not transparent.
Shedding more light on this, Dr. Zulfiqar Ali Shah, stated:
There is a difference of opinion among jurists as regards wiping over socks made of cloth. Many contemporary jurists permit wiping over cloth socks if they were put on in the state of wudu’ because cleanliness and purity can be maintained through the socks as they are usually supplemented with leather shoes or the body.
Hanafi jurists, however, require that the socks must be thick enough so that the water does not seep through them.
To make things easy for our brothers and sisters who are working among non-Muslims in offices and traveling through airports where security concerns are heightened, it is permissible to wipe over socks even though they are not thick like leather socks. Allah the Almighty wants to make things easy, not difficult.