Is it permissible for a mother to give her children from her zakat money to register a house for them or to establish a business, when their father already spends on them what suffices their needs?

It is not permissible for a mother to give her Zakat to her children, except in the event that the father is unable to provide for them. If the father is providing for them, then they are considered wealthy, and it is not permissible for the mother or anyone else to give them Zakat. The mother must disburse her Zakat to the categories specified in the Holy Qur’an, which are: the poor, the needy, those employed to collect it, those whose hearts are to be reconciled, for freeing captives, those in debt, in the cause of Allah, and the wayfarer.