Quotient-powering-invariant subgroup

Definition
A normal subgroup $$H$$ of a group $$G$$ is termed a quotient-powering-invariant subgroup if, for any prime number $$p$$ such that $$G$$ is a powered for $$p$$, the quotient group $$G/H$$ is also powered for $$p$$.

Properties whose conjunction with powering-invariance implies quotient-powering-invariance
The relevant subgroup property is normal subgroup satisfying the subgroup-to-quotient powering-invariance implication. In fact, the conjunction of this with powering-invariant subgroup precisely gives quotient-powering-invariant subgroup.

This property is implied both by being a central subgroup and by being a normal subgroup contained in the hypercenter.