Let $G$ be a group with generating set $S$ and $\Gamma$ the corresponding Cayley graph. Let $Aut(\Gamma)$ be the automorphism group of the corresponding undirected graph and $Aut^+(\Gamma)$ the subgroup preserving the directions and labellings of the Cayley graph.
It can happen that $Aut^+(\Gamma)$ is strictly smaller than the full automorphism group, for example with $G=S_3$ and $S$ consisting of $(12)$ and $(123)$.
I am looking for an example where the full automorphism group is much larger is the sense that the index $[Aut(\Gamma):Aut^+(\Gamma)]$ is infinite.