Is it permissible for me to take back my wife after discovering her talking to a non-mahram man on "Facebook," knowing that my mother refuses that and refuses to enter my house if I take her back?

A wife's conversation with a foreign man via Facebook is a reprehensible act and a dereliction of her duty towards Allah and her husband. The wife must protect her husband in his absence, as stated in Surat An-Nisa. If the wife repents and becomes upright, there is no harm in her return. If the mother is not pleased, an effort must be made to gain her satisfaction. If the wife persists in her state, then obedience to the mother is obligatory, and the wife should not be taken back. In the event of divorce, custody belongs to the mother unless she remarries or there is an impediment, in which case custody transfers to the next most deserving person, such as the maternal grandmother.