Given a simple, undirected graph and a vertex $v$ of the graph, let $L_v$ denote the set of automorphisms of the graph that fixes the vertex $v$ and each of its neighbors. When the graph is vertex-transitive, the vertex-neighborhood stabilizer $L_v$ is independent of the choice of $v$.
I noticed that for many vertex-transitive graphs, $L_v$ happens to be either trivial or is isomorphic to the direct product of copies of $C_2$. Thus $L_v$ happens to be isomorphic to $C_2^k$ for some $k \ge 0$. For example, for the modified bubble-sort graph on 24 vertices, $L_v$ is the Klein four-group $C_2 \times C_2$, and for many Cayley graphs generated by transposition sets, $L_v$ is trivial (so $k=0$ in this case) (cf. http://arxiv.org/abs/1205.5199). For the complete transposition graph, $L_v \cong C_2$ (cf. http://arxiv.org/abs/1404.7363). I also considered the Petersen graph, and again $L_v \cong C_2$. Is this a coincidence or is there some result that says that for some families of vertex-transitive graphs or for certain families of Cayley graphs, $L_v \cong C_2^k$ for some $k$? What are some counterexamples - for example, what are some (vertex-transitive) graphs for which $L_v \cong C_3$, say?