Is it obligatory for a woman to keep her mother's family name if she does not know her father's family name, and does she change her family name when she changes her first name?

It is forbidden for a person to ascribe himself to other than his father or to a people he does not belong to. The Prophet, peace and blessings be upon him, said: "No man claims to be the son of other than his father, while he knows it, except that he has committed an act of disbelief." And he said: "Whoever claims a people to whom he has no lineage." Many evils result from this false attribution.

She must change her name. If she does not know her father's name or his family, she should name herself with a general name such as: Fatima bint Abdullah or Fatima bint Abd al-Rahman.