Let $G$ be some (infinite) group, and let $Aut(G)$ be its automorphism group. Assume $H\leq Aut(G)$.
Under what conditions can I construct another group (or, say, graph), $\hat{G}$, such that $Aut(\hat{G})=H$? If so, is there an algorithm to do so?
I am pretty sure that this is not always possible - there are some groups which never occur as automorphism groups of other groups. I would therefore be interested to know if we can apply conditions on either $G$ or $H$ to get this to work.
I presume $H\lhd G$ is not sufficient, as then if $H$ is a centerless group which never occurs as an automorphism group of another group then we would have a counter-example (as $ Inn(G) \cong G $ if the centre of $G$ is trivial). So...what about characteristic?