Connected arc-transitive graph

Question

Let $\Gamma$ be a connected arc-transitive graph and let $G=Aut(\Gamma)$. Let $x,y \in V(\Gamma)$ be such that $ e=\{ x,y \} \in E(\Gamma)$. Let $G_x$ and $G_e$ be stabilisers in $G $ of the $x$ and $e$. Let $H$ be the subgroup of $G$ is generated by vertex and edge stabilisers. If $\gamma =(x, x_1, x_2, \ldots ,x_n)$ is a path in $\Gamma$, prove that there is $h \in H$ such that h(x)=$x_n$