Let be $V\leq \mathrm{Aut}\left( G\right)$, $N\vartriangleleft G$ and $V$-invariant. Consider the semidirect product $G\rtimes V$.
Let $X$ be the preimage of $C_{G/N}\left( V\right) =\left\{ gN~;~g^{v}N=gN,\forall v\in V\right\} $ in $G$. Is it true that $X$ normalizes $NV$?