What should a husband do with his disobedient wife who spends most of her time outside the home, incited by her family?

A wife must obey her husband in what is right. Her disobedience to his commands is considered nushuz (rebellion), which warrants advice, then boycotting in bed, then non-violent striking, as stated in the Almighty's قول (statement): (As for those [wives] from whom you fear nushuz, admonish them, and abandon them in bed, and strike them. But if they obey you, seek no means against them. Indeed, Allah is ever Exalted and Grand) [An-Nisa:34].

The matter should be handled gently and leniently by clarifying the obligation of obedience and the punishment for nushuz, and by reassuring her that she will not be completely prevented from seeing her family, but rather that the relationship with them will be organized. If her family incites her to disobedience, then he has the right to prevent her from going to them or visiting them.