Let $G$ be an extraspecial group of order $p^{2r+1}$ (where $p$ is an odd prime), and let $V$ be a faithful representation of $G$. Consider the normal subgroup $H$ of $Aut(G)$ consisting of all elements of $Aut(G)$ acting trivially on the center $Z(G)$.

Is every element of $H$ induced by $SL(V)$ ? More generally, is it true that $H=Aut_{SL(V)}(G)$ ?