Derivative Citizenship is possible when a mother who is a United States citizen has a child abroad under certain circumstances, depending on when the child was born and when the mother left the United States. The child may be able to obtain their Citizenship derived from the mother. In certain circumstances, it would depend on when the mother lived in the United States and whether she satisfied the time requirements.
If she did not satisfy the time requirements to qualify for Derivative Citizenship, she would have to petition the child through a general family petition called N600 procedures.