What is the ruling on a wife who puts powder on her face when leaving the house and refuses to stop doing so, claiming that all women do more than this, and threatens divorce when she is forbidden from it? Is the husband held accountable for his silence, and what is the solution in this situation?

A woman is not permitted to go out adorned in a way that non-mahram men can see her. If treating the wife's defiance (nushuz) is unsuccessful, then there is no objection to divorcing her, or to enduring her. Divorce is recommended if the woman neglects her obligatory duties towards Allah, such as prayer, and cannot be compelled [to perform them], or if she is unchaste, because keeping her would diminish religious commitment.