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](https://mathoverflow.net/users/22989/jeremy-rickard) for the correction). Are there any other examples?



When is Aut(G) the symmetric group of an Aut(G)-invariant generating set?
For $G$ abelian, the complete list is $C\_2$, $C\_3$, $C\_4$, $C\_6$ and $C\_2\times C\_2$.


Proof:
Suppose that $X$ contains a non-involution $x$. Then $x^{-1}\in X$ (since inversion in an automorphism of $G$) but then $\{x,x^{-1}\}$ is a block of size 2 for the action of Aut(G) on $X$ and thus $|X|=2$ and $X=\{x,x^{-1}\}$ and $G$ is cyclic. Since $X$ is $Aut(G)$-invariant, $G$ must have only two element of order $|G|$ and this happens only when $G$ is $C\_3$, $C\_4$ or $C\_6$. We may thus assume that $G$ is generated by involutions and hence is an elementary abelian $2$-group. It's easy to check that $C\_2^n$ has this property only for $n=1$ and $n=2$.


(I was assuming $G$ is finite, but the proof can probably be adapted.)


For $G$ nonabelian, I think the only example is $G=Sym(3)$ with $X$ the set of involutions. You could probably prove this following YCor's comments for example.


