# Automorphism groups of vertex transitive graphs

Does there exist a finite nonoriented graph whose automorphism group is transitive but not generously transitive (that is, it is not true that each pair $(x,y)$ of vertices can be interchanged by some automorphism)?

I know, for example, that Cayley graphs are always generously vertex-transitive.

In particular if $G$ is not abelian and $X$ is a so-called graphical regular representation (abbreviated GRR) for $G$, then its automorphism group is not generously transitive. The key property of a GRR is that the stabilizer of a vertex is trivial.