Is a judge permitted to divorce a wife if she requests a khul' due to lack of kind treatment and extreme anger, and the husband refuses to divorce her?

The religiously mandated duty is for the husband to treat his wife with kindness. If the lack of kind treatment constitutes clear harm, such as verbal abuse or severe physical beating, then this is a reason that permits divorce without compensation and without the husband's consent, provided the harm is proven by two witnesses.

However, if the situation does not reach the level of clear harm, but the wife can no longer tolerate staying and fears she will not fulfill her duties to Allah, then she is permitted to request a divorce or a khul' (divorce initiated by the wife) with compensation, and it is recommended for the husband to accept it. If the husband refuses the khul', he is not compelled to accept it according to most scholars, but some scholars lean towards compelling him.