Finding automorphism groups of double coset graphs and Cayley graphs are equivalent

It has long been known that a vertex-transitive graph $\Gamma$ is isomorphic to a double coset graph $\text{Cos}(G,H,S)$ of a transitive group $G\le\text{Aut}(\Gamma)$, a vertex stabilizer $H\le G$, and some subset $S\subseteq G$. We show that the automorphism group of the Cayley graph $\text{Cay}(G,S)$ with connection set $S$ can be obtained from the automorphism group of $\text{Cos}(G,H,S)$ and vice versa. We also show that the isomorphism problem for double coset graphs is equivalent to the isomorphism problem for Cayley graphs provided one knows all groups $G$ for which a fixed Cayley graph is a Cayley graph of $G$. Our main tool is a "recognition theorem", which recognizes when a Cayley graph of a group $G$ is a wreath product of two graphs based upon its connection set.