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 haraam; it is also a form of falsehood, and humiliation of the woman. Anyone who has done this must repent to Allaah 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).