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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.

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.

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.

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.

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

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.