Given vertex transitive $G$ and $H$ such that $|\mathcal{V}(G)|<|\mathcal{V}(H)|$ and $\mathcal{Aut}(G)\supset\mathcal{Aut}(H)$, is it true $G\leq H$? Conversely if $G\leq H$, what can we say about $\mathcal{Aut}(G)$ and $\mathcal{Aut}(H)$?