Commutator-in-center subgroup

Definition
A subgroup $$H$$ of a group $$G$$ is termed a commutator-in-center subgroup if it satisfies the following equivalent conditions:


 * 1) $$[G,H] \le Z(H)$$ where $$[G,H]$$ denotes the commutator of two subgroups and $$Z(H)$$ denotes the center of $$H$$.
 * 2) $$H$$ is a normal subgroup of $$G$$ and $$G,H],H]$ is trivial, i.e., $H$ is a [[commutator-in-centralizer subgroup.

A group has this property as a subgroup of itself if and only if it has nilpotency class two.

Related group properties
A group $$H$$ has the property that for every group $$G$$ containing $$H$$ as a normal subgroup, $$H$$ is a commutator-in-center subgroup of $$G$$, if and only if $$H$$ is a group whose inner automorphism group is central in automorphism group.