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Let $G$ be a finite group and let $\sigma$ be an automorphism of $G$. Generally, if $H$ is a subgroup of $G$, then $\sigma|_{H}$ ($\sigma$ restricted to $H$), is not always an automorphism of $H$.

My questions are:

If $\sigma\neq id$ and has the property that $\sigma|_{H}$ is an automorphism of $H$ for every subgroup $H$, what can be said about $G$ and what can be said about $\sigma$ itself? Can it be an inner automorphism? Do we have always at least one such non-trivial automorphism?

I tried to answer this questions on groups that generated by two elements and didn't go far...

Thanks!

boaz
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    Maybe a good place to start looking is $G=S^n$, as all automorphisms of $S^n$ are inner (if $n \geq 7$ I think). Then for every $\sigma \neq \mathrm{id}$, there is a subgroup $\langle (ij)\rangle$ not preserved by $\sigma$ (as $\sigma(ij)=(ij)$ will imply that $\sigma$ is the identity). – Nicolás Vilches Aug 15 '21 at 14:10

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