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 $\mathcal{Aut}(G)\supset\mathcal{Aut}(H)$?


(I suspect the answer to the second question is no.)


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 $\mathcal{Aut}(G)\supset\mathcal{Aut}(H)$?


(Again I suspect the answer to the second question is no.)


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



On homomorphisms between vertex transitive graphs
The answer to the first question is certainly no. The automorphism group of the Petersen graph $H$ is $\mathfrak S\_5$ so is equal to the automorphism group of the complete graph $G=K\_5$, which has strictly fewer vertices. Both these graphs are vertex transitive and yet the Petersen graph has no $5$-clique, so $G$ is not a subgraph of $H$.


The second question is very strange. I must misunderstand, otherwise the isolated vertex in a non-trivial vertex transitive graph (say a 3-cycle for concreteness) is an immediate counterexample.


The third question also immediately admits a negative answer: take $G$ equal to the Petersen graph again and $H$ equal to 5 isolated vertices.


I don't know about the fourth question, though one can reduce to $H$ being a (connected) core.


