Universal power not implies class-preserving

Statement
A universal power automorphism of a group (i.e., an automorphism obtained by taking the $$n^{th}$$ power for some integer $$n$$) need not be a class-preserving automorphism: it need not preserve conjugacy classes of elements.

Stronger facts

 * Universal power not implies center-fixing
 * Universal power not implies IA

Corollaries

 * Subgroup-conjugating not implies class-preserving
 * Normal not implies class-preserving

Example of an Abelian group
Consider the cyclic group of order $$p$$, where $$p$$ is an odd prime. The inverse map is a universal power automorphism of this group, and is not the identity map. Since the group is Abelian, the conjugacy classes are singleton, so we have a universal power automorphism that is not class-preserving.

The example generalizes to any Abelian group whose exponent is bigger than two.