Local powering-invariant subgroup

Definition
A subgroup $$H$$ of a group $$G$$ is termed a local powering-invariant subgroup if the following hold:


 * 1) Whenever $$h \in H$$ and $$n \in \mathbb{N}$$ are such that there is a unique $$x \in G$$ such that $$x^n = h$$, we must have $$x \in H$$.
 * 2) Whenever $$h \in H$$ and $$p$$ is a prime number such that there is a unique $$x \in G$$ such that $$x^p = h$$, we must have $$x \in H$$.

Facts

 * Center is local powering-invariant
 * Derived subgroup not is local powering-invariant