Let:
- $G$ be a group;
- $H\le G$;
- $\theta,\hat\theta$ be Cayley embeddings.
Do there exist embeddings $\epsilon:H \to G$ and $\hat\epsilon:{\rm Sym}(H) \to {\rm Sym}(G)$ such that the above diagram commutes; i.e. such that $\hat\epsilon\hat\theta=\theta\epsilon$?
The only I can think of for $\epsilon$ is either the natural one ($\epsilon(h)=h$) or by conjugacy ($\epsilon(h)=ghg^{-1}$), while for $\hat\epsilon(\sigma)$ is the extension of $\sigma \in \operatorname{Sym}(H)$ by the identity map of $G\setminus H$, but I didn't come to a conclusion with them.