If $G$ is a group and $S$ is a generating set for it, then we can define a distance metric based upon the minimum $S$ operations needed to move from one element of $G$ to another. As a result, $G$ can be shown to be a subgroup of Isom$(G)$. (Prove than an arbitrary group G with the following metric can be realised as a subgroup of Isom(G) shows this).
However, every example I can see has $G$ isomorphic to Isom$(G)$. Is there an example where $G$ is a strict subgroup of Isom$(G)$?