IA not implies class-preserving

This article gives the statement and possibly, proof, of a non-implication relation between two automorphism properties. That is, it states that every automorphism satisfying the first automorphism property (i.e., IA-automorphism) need not satisfy the second automorphism property (i.e., class-preserving automorphism)
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Statement

Property-theoretic statement

The automorphism property of being an IA-automorphism does not imply the automorphism property of being a class-preserving automorphism.

Verbal statement

There exists a group with an IA-automorphism (an automorphism that is identity on the Abelianization) that does not send every element to within its conjugacy class.

Proof

An example of a perfect group

If $G$ is a perfect group, then every automorphism of it is an IA-automorphism, so all we need to do is exhibit an automorphism of $G$ that is not class-preserving. Indeed, take $G$ to be the alternating group of degree five, acting on $\{1,2,3,4,5\}$. Consider the automorphism of $G$ induced via conjugation by any odd permutation. We claim that such an automorphism sends the five-cycle $(1,2,3,4,5)$ to an element outside its conjugacy class.

Suppose not. Then, there exists an even permutation that has the same effect by conjugation on $(1,2,3,4,5)$. Taking their ratio, we get an odd permutation that commutes with $(1,2,3,4,5)$. But we know that the only permutations that commute with $(1,2,3,4,5)$ are its powers, which are all even permutations -- hence a contradiction.

A more generic way of saying this is that if $G$ is a perfect centerless group that is not conjugacy-closed in the automorphism group, then $G$ has IA-automorphisms that are not class-preserving.