Let $G$ be a group, $X$ a generating set of $G$. Suppose $X$ is $\operatorname{Aut}(G)$-invariant, i.e. $\sigma(X)\subseteq X$ for all $\sigma \in \operatorname{Aut}(G)$. When is the restriction homomorphism $$ \begin{gathered} \operatorname{Aut}(G) \to \operatorname{Sym}(X)\\ \sigma \mapsto \sigma|_X \end{gathered}$$ an isomorphism?

**Example:** ~~$A_4$ with $X=\{\text{3-cycles}\}$ satisfies the above properties.~~ $C_2\times C_2$ works, and so do all cyclic groups of order up to 4 (Thanks Jeremy Rickard for the correction). Are there any other examples?

2more comments