In general there is no relation between automorphism groups of subgraphs and the main graph. However, this question is about vertex transitive graphs.

Given vertex transitive $G$ and $H$ such that $|\mathcal{V}(G)|<|\mathcal{V}(H)|$.

If $\mathcal{Aut}(G)\supset\mathcal{Aut}(H)$, is $G\leq H$?

If $G\leq H$, is $\mathcal{Aut}(G)\supset\mathcal{Aut}(H)$?

Given vertex transitive $G$ and $H$ such that $|\mathcal{V}(G)|>|\mathcal{V}(H)|$.

If $\mathcal{Aut}(G)\supset\mathcal{Aut}(H)$, is $G\rightarrow H$?

If $G\rightarrow H$, is $\mathcal{Aut}(G)\supset\mathcal{Aut}(H)$?

$\rightarrow$ implies homomorphism exists in the direction suggested.