For a given finite group $Г$ , determine an infinite number of mutally nonisomorphic graphs whose groups are isomorphic to $Г$. For a given finite group $Г$ , determine an infinite number of mutally nonisomorphic graphs whose groups are isomorphic to $Г$.
I know that $Г$ is generated by $\Delta$ and for any finite $Г$, there exist a graph $G$ such that $Aut(G) \cong Г$
But how does these info can help me find the number of mutally nonisomorphic graphs whose groups are isomorphic to $Г$?

Given a group $\Gamma = \{g_1,\ldots,g_n\}$, construct an edge-colored digraph on vertex set $\Gamma$, and with arcs $(g_i,g_j)$ having color $g_k$ iff $g_k=g_i g_j^{-1}$.  Then the edge-colored digraph has automorphism group $R(\Gamma)$, the right regular representation of $\Gamma$, which is isomorphic to $\Gamma$. There are an infinite number of ways to replace this edge-colored digraph with a simple, undirected graph that has the same automorphism group as the edge-colored digraph.  This yields an infinite number of undirected graphs whose groups are isomorphic to $\Gamma$.  
