Powering-invariant normal subgroup of nilpotent group

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


 * 1) $$G$$ is a nilpotent group and $$H$$ is a powering-invariant normal subgroup of $$G$$, i.e., $$H$$ is a normal subgroup and is a powering-invariant subgroup of $$G$$. Here, by powering-invariant, we mean if $$p$$ is a prime number such that $$G$$ is $$p$$-powered, $$H$$ is also $$p$$-powered.
 * 2) $$G$$ is a nilpotent group and $$H$$ is a quotient-powering-invariant subgroup of $$G$$, i.e., $$H$$ is a normal subgroup and if $$p$$ is a prime number such that $$G$$ is $$p$$-powered, the quotient group $$G/H$$ is also $$p$$-powered.

Equivalence of definitions

 * (1) implies (2) follows from normal subgroup of nilpotent group satisfies the subgroup-to-quotient powering-invariance implication.
 * (2) implies (1) follows from quotient-powering-invariant implies powering-invariant.