Let $H$ be a group. Can $H$ be embedded into a bigger group $G$ such that $H\triangleleft G$ and for every $\varphi\in\text{Aut}(H)$ there exists $g_\varphi\in G$ with $\varphi(h)=g_\varphi hg_\varphi^{-1}$ for all $h\in H$? If so, can it be done in some canonical, or even in some sense minimal way?
This question came up while working on Sylow-groups, but it isn't of any actual use there. I just wondered :)