Group whose inner automorphism group is central in automorphism group

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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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Definition

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

Formalisms

In terms of the supergroup property collapse operator

This group property can be defined in terms of the collapse of two subgroup properties in the following sense. Whenever the given group is embedded as a subgroup satisfying the first subgroup property (normal subgroup), in some bigger group, it also satisfies the second subgroup property (commutator-in-center subgroup), and vice versa.
View other group properties obtained in this way

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 $G$. Further information: normal subgroup whose inner automorphism group is central in automorphism group is commutator-in-center

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Abelian group	inner automorphism group is trivial			|FULL LIST, MORE INFO
Group whose automorphism group is abelian	automorphism group is abelian			|FULL LIST, MORE INFO
Cyclic group	generated by one element	(via abelian, also via abelian automorphism group)		Group whose automorphism group is abelian|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Group of nilpotency class two	inner automorphism group is abelian			|FULL LIST, MORE INFO