Normal subgroup satisfying the subgroup-to-quotient powering-invariance implication

Definition
Suppose $$H$$ is a normal subgroup of a group $$G$$. We say that $$H$$ satisfies a subgroup-to-quotient powering-invariance implication if for any prime $$p$$ such that both $$G$$ and $$H$$ are powered over $$p$$, the quotient group $$G/H$$ is also powered over $$p$$.