Suppose $G = A \rtimes B$ is a group, where $B$ acts faithfully on $A.$ Since B acts faithfully on A, $Z(G) \subseteq A.$ If $z \in Z(G),$ then what can we say about the action of $B$ on $\displaystyle \frac{A}{\langle z \rangle}?$ Will it also be faithful?
I can only see that if suppose the action is conjugation, then some $b \in B$ will lie in the kernel of its action on $\displaystyle \frac{A}{\langle z \rangle},$ if and only if $(a,b) \in \langle z \rangle \ \forall \ a \in A.$ I am really stuck in proceeding further. Do we have some general result in this direction?
In general the action of $B$ on $A/\langle z\rangle$ is not faithful. Consider $A=C_4$, $B=C_2$ such that $G\cong D_8$.