Is it permissible to relinquish half of the inheritance (the father's house) to the husband as common property upon receiving the price for this relinquishment, and is it permissible for the husband to build in part of the house without consulting the other heirs, and is it possible to retract this relinquishment?

In this case, the relinquishment is a gift (hibah), not a sale, because it is without consideration. What actually transpired is what counts, and the legal professionals in the questioner's country should be consulted regarding its legal implications. Scholars have differed on the gifting of undivided common property (mushā‘). The majority stipulates possession (ḥiyāzah) for it to be valid, while the Hanafis permitted it if it is something that can be divided. The Shafi'is and Hanbalis held that possession of a common share is achieved by possessing the whole. As for the Malikis, they are of the view that possession of a common share is achieved by taking control of it just as its owner had control over it with his partner.

Accordingly, the questioner is not permitted to revoke her gift if the husband has taken possession of it through قبض (qabḍ). Otherwise, she may revoke it. If the gift becomes binding, the husband is not allowed to construct or demolish anything in the inherited house except with the permission of the remaining heirs. However, he may demand partition, and the remaining heirs are obligated to comply with his demand.