Class-preserving implies IA

Property-theoretic statement
The automorphism property of being a fact about::class-preserving automorphism is stronger than the automorphism property of being an fact about::IA-automorphism.

Class automorphism
An automorphism of a group is termed a class automorphism if it sends every element to an element in its conjugacy class; in other words, it preserves conjugacy classes.

IA-automorphism
An automorphism of a group is termed an IA-automorphism if it induces the identity map on the Abelianization of the group. In other words, it sends every coset of the commutator subgroup, to itself.