A power automorphism maps every subgroup of a group to within itself, with equality if the group is finite. More specifically, for a subgroup $H$ of $G$, a power automorphism $f$ has $f(H) \subseteq H$. If $G$ is finite then $f(H) = H$.
There are lots of these of the form $f:G \to G$, $f(x) = x^n$, especially for finite abelian groups. This is a universal power automorphism.
I want to find an example of a non-trivial power automorphism of a finite group that isn't universal, particularly with some insight in how it is constructed. Can someone give me an example?
Here's an example: consider the automorphism of the quaternion group $Q_8$ whose action on its generators $i$, $j$, $k$ is as follows: $$i \mapsto i \hspace{10pt} j \mapsto -j \hspace{10pt} k \mapsto -k$$
(Of course, this also works for $i \mapsto -i , j \mapsto j , k \mapsto -k$ and $i \mapsto -i \hspace{10pt} j \mapsto -j \hspace{10pt} k \mapsto k$.)