Is it permissible to keep one's wife and abandon her in bed for the sake of raising the children? And is it permissible to expose her after she was found with her colleague in the car at night and swore that she repented and no adultery occurred?

It is not permissible to accuse one's wife of adultery or to expose her. The husband must either divorce her or live with her honorably. If the wife repents and shows no further suspicion, then he should keep her and live with her honorably, having a good opinion of her, and not dwelling on the past, for repentance erases what came before it.