IA not implies class-preserving

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.

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.