Quotient-powering-faithful subgroup

Definition
Suppose $$G$$ is a group and $$H$$ is a normal subgroup. We say that $$H$$ is quotient-powering-faithful in $$G$$ if the following holds: for any prime number $$p$$ such that $$G/H$$ is $$p$$-powered, so is the whole group $$G$$.

Facts

 * Derived subgroup is quotient-powering-faithful in nilpotent group