Let $\Gamma=Cay(G,S)$ be a Cayley graph over a group $G$, $H$ be a proper subgroup of $G$ and $\Sigma=Cay(H,T)$ where $S$ and $T$ are inversed-closed subsets of $G$ and $H$ not containing idendity, respectively. Is there any relation between $Aut(\Gamma)$ and $Aut(\Sigma)$ in general? When $T=H\cap S$, what can we say?



Is there any relation between automorphism group of a Cayley graph over a group and over its subgroup?
I am reasonably confident that the real answer to your question is "No". For example,
if $T=H\setminus 1$ then the automorphism group of $\mathrm{Cay}(H,T)$ is the full
symmetric group. For another, take $S=T$; then $\Gamma$ will have a large automorphism
graph, a wreath product. (If you want your Cayley graphs connected, take complements.)
So adding generators may reduce symmetry, or increase it.


Note that questions phrased as you have are a bit difficult to answer, since the
answer depends on what you mean by "any relation between".


