Automorphism Group of Simple Graph

Question

Suppose $G$ is a group. Does there always exist a simple graph $\Gamma$ such that $Aut(\Gamma)=G$, i. e. such that $G$ is the automorphism group of $\Gamma$?

Given this I'd say no, but I would really appreciate if someone could help figure out an example.

Answer 1

I did some Googling and found Frucht's Theorem. The argument seems to involve modifying the Cayley graph of $G$ to remove colors and orientations while not introducing extra automorphisms.