It is not permissible for a woman to go out of her husband’s house without permission, rather many of the scholars regarded this as nushooz (defiance) and going against the husband, if there is no excuse for doing that, such as if her husband is harming her in a manner that she cannot ward off and so on.
Moreover, if the wife is withholding herself from her husband, he is no longer obliged to spend on her because of her defiance, as the fuqaha (jurists) have stated. See al-Mughni, 8/182.
What your brother should do is to handle the situation wisely and carefully in order to bring his wife back home. He should remind her of Allah, and remind her family of Him; if he cannot do that himself, then he should enlist the help of some relatives who have knowledge, experience and wisdom, and get them involved so that they can solve the issue.
He should exercise deliberation and not be hasty in taking decisions, for “Deliberation is from Allah and haste is from the Shaytan,” as the Prophet (peace and blessings of Allah be upon him) said. (Classed as saheeh by al-Albani in al-Silsilah al-Saheehah, 1895).
A man may make a decision at the time of anger, then regret it, but at a time when regret will be to no avail.
He should also adopt an attitude of patience, put up with his wife and try to put an end to the disputes between them that have gone on for years and years. Let him start a new life with her, forgetting the past and its arguments.
No one is perfect, so he should accept her good points and overlook her bad points, and try to change her in a wise and calm manner. The Prophet (peace and blessings of Allah be upon him) said: “No believing man should hate a believing woman, for if he dislikes one of her characteristics, he will be pleased with another.”
This means that he should not hate her, because if he sees in her a characteristic that he dislikes, he will find another that is pleasing; she may be bad-tempered but at the same time religiously-committed, or beautiful, or chaste, or kind to him, and so on. End quote.
This is how all people are; they have good qualities and bad qualities. The wise man is the one who strikes the right balance between good and bad, accepting the good things and overlooking the bad, whilst also trying to correct them.
If the husband does all that but the woman still does not change, then he may refer to the shar’i courts to resolve this dispute.
And Allah is the One Whom we ask to set the affairs of all the Muslims straight.
And Allah knows best.