Powering-invariant normal subgroup

Definition
A subgroup $$H$$ of a group $$G$$ is termed a powering-invariant normal subgroup if it is both a powering-invariant subgroup and a normal subgroup of the whole group. Here, powering-invariant means that for any prime number $$p$$ such that $$G$$ is powered over $$p$$, we have that $$H$$ is also powered over $$p$$.