Question: Let $G$ be a finite group. If the group automorphism $\sigma$ satisfies that $\sigma(H)=H$ for some nontrivial subgroup $H$, we call $\sigma$ has the preserving property.
Now, let $G$ be a non-abelian group. How to prove that any automorphism has the preserving property?
Trying: I have tried the abelian case, once we decomposite the abelian group into the product of cyclic groups with distinct prime powers, we can construct such automorphism by shifting each cyclic group. But I have no idea with the non-abelian case.