Is it permissible to give Zakat to a female relative who is married to an unemployed man, while she is employed and has four children, two of whom work with her, and the eldest of them is 17 years old?

If your relative's husband is in financial difficulty, and she and her children are poor, it is permissible to give her from the Zakat. Otherwise, it is not permissible, because she has sufficient means through the maintenance she is entitled to.