PARIS — Having spent the last five years living in Syria, where she joined Islamic extremists, Emilie König, 33, wants to come home to France. But does France want her back?

The daughter of a policeman from a small town in Brittany, she converted to Islam as a teenager. After she began covering herself from head to toe in a black abaya and veil, she felt so scorned in France that she left her two small children to go to Syria, eventually becoming a prominent propagandist and recruiter for the Islamic State.

“She would like to come back; she has asked for pardon from her family, her friends, her country,” her mother said in an interview with Ouest-France, a newspaper in Brittany, after speaking to her daughter by telephone two weeks ago.

Ms. König’s personal story is unusual, not least because she is a convert and gained prominence within the male-dominated Islamic State. Yet the quandary her case poses is an increasingly common one for France and other European countries: What should they do when citizens who are former Islamic State fighters or supporters want to return?