For a subgroup $H$ contained in the center of a finite group $G$ , $F(G/H)=F(G)/H$, where $F$ denotes Fitting subgroup

Question

For any group $G$, let $F(G)$ denote the Fitting subgroup. Now if $G$ is a finite group and $H$ is a subgroup of $Z(G)$, then how to show that $F(G/H)=F(G)/H$ ?

Please help.

Answer 1

The advice of prof. Holt is very valid, but let me give you a small hint: put $F(G/H)=K/H$. Then $K/H$ is nilpotent and $H \subseteq Z(G)$ implies $H \subseteq Z(K)$. What does this say about $K/Z(K)$?