Powering-invariant central subgroup

Definition
A subgroup $$H$$ of a group $$G$$ is termed a powering-invariant central subgroup if it satisfies the following equivalent conditions:


 * 1) $$H$$ is both a powering-invariant subgroup of $$G$$ and a central subgroup of $$G$$.
 * 2) $$H$$ is both a quotient-powering-invariant subgroup of $$G$$ and a central subgroup of $$G$$.