If we have a short exact sequence $1 \rightarrow C \rightarrow G \rightarrow Q \rightarrow 1$ where $C$ is central in $G$ and $Q \cong G/C$, how can I find an isomorphism between $Hom(Q,C)$ (which is a group under pointwise multiplication) and the automorphisms of $G$ which act trivially on $C$ and $Q$?
I don't know any cohomology, so any hints from a more elementary perspective would be appreciated.