Quotient-local powering-invariant subgroup

Definition
Suppose $$G$$ is a group and $$H$$ is a normal subgroup of $$G$$. Let $$\varphi:G \to G/H$$ be the corresponding quotient map. We say that $$H$$ is a quotient-local powering-invariant subgroup of $$G$$ if the following equivalent conditions hold:


 * 1) For any $$g \in G$$ and any natural number $$n$$ such that there exists a unique $$x \in G$$ satisfying $$x^n = g$$, we also have that $$\varphi(x)$$ is the unique element of $$G/H$$ whose $$n^{th}$$ power is $$\varphi(g)$$.
 * 2) For any $$g \in G$$ and any prime number $$p$$ such that there exists a unique $$x \in G$$ satisfying $$x^p = g$$, we also have that $$\varphi(x)$$ is the unique element of $$G/H$$ whose $$p^{th}$$ power is $$\varphi(g)$$.