Group whose inner automorphism group is central in automorphism group

Definition
A group whose inner automorphism group is central in automorphism group is a group whose defining ingredient::inner automorphism group is a defining ingredient::central subgroup of the defining ingredient::automorphism group. In other words, every defining ingredient::inner automorphism of the group is a defining ingredient::central automorphism.

Formalisms
A group $$H$$ is a group whose inner automorphism group is central in automorphism group iff, for any group $$G$$ containing $$H$$ as a normal subgroup, $$H$$ is a commutator-in-center subgroup of $$G$$.