Is it permissible to give zakat to a poor sister whose husband does not earn enough for his family, and who has a young son who has not completed his studies and is not working, knowing that the sister has her own inheritance that she can sell, and that the zakat will be given directly to the sister for her to spend on her family's behalf?

It is not permissible to give your sister Zakat because she is not poor, as her husband suffices her. Even if he were poor, she has money from an inheritance that meets her needs. Allah Almighty has clarified the categories for Zakat distribution in His saying: "Indeed, [prescribed] charitable donations are only to be given to the poor and the needy, and those who work on [collecting] them, and those whose hearts are to be reconciled, and to [free] captives, and to those in debt, and for the cause of Allah, and for the stranded traveller—an obligation from Allah. And Allah is All-Knowing, All-Wise."