My sister is married to a man who only works sometimes but he is also stingy and does not spend on her or her children. She works and pays for all the household expenses, and now she is being forced to complete her studies so that she can get a higher certificate, otherwise she will lose her job, but she needs money to do these studies. Is it permissible for me to give the zakaah of my wealth to her?.
It is permissible to give zakaah to a seeker of worldly knowledge who does not have enough to support him, if the knowledge that he is studying is permissible and is needed, so that he will be able to work and find a job, because getting these certificates has become a real necessity without which one cannot usually find employment.
Al-Mardaawi says in al-Insaaf (3/218): Shaykh Taqiy al-Deen favoured the view that it is permissible to take zakaah in order to buy books that he needs including books of knowledge that is indispensable for his religious and worldly affairs. End quote. And this is the correct view. End quote.
Based on this, there is nothing wrong with your giving the zakaah of your wealth to your sister.
And Allaah knows best.