It is haram for a wife to take her husband’s name
Question

Many sisters in our community have taken the names of their husbands without realizing that they should have kept their father's name. Should they change their name back to the father's name or is it allowed to keep the husband's name?
Also, if a person is born out-of-wedlock, should they carry their father's last name or their mother's last name? What is the evidence for this?
Praise be to Allah.It is not permitted for anyone to claim to belong to anyone other than his father. Imitating the kuffaar by dropping the wifes surname and giving her the husbands name is haram; it is also a form of falsehood, and humiliation of the woman. Anyone who has done this must repent to Allah and put it right by going back to her fathers name.
With regard to a child born out of wedlock, he should be given his mothers name and cannot be given the name of the adulterer. (For more information, see Question #1942 and 284).