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!