Local powering-invariant subgroup containing the center is intermediately powering-invariant in nilpotent group

Statement
Suppose $$G$$ is a nilpotent group and $$H$$ is a subgroup containing the center of $$G$$ that is also a local powering-invariant subgroup of $$G$$. Then, $$H$$ is an intermediately powering-invariant subgroup of $$G$$. Explicitly, suppose $$K$$ is a subgroup of $$G$$ containing $$H$$. Then, $$H$$ is a powering-invariant subgroup of $$K$$.

Related facts

 * Local powering-invariant subgroup containing the center is intermediately local powering-invariant in nilpotent group

Facts used

 * 1) Torsion-freeness for a prime is subgroup-closed
 * 2) uses::Equivalence of definitions of nilpotent group that is torsion-free for a set of primes

Proof
Given: A nilpotent group $$G$$, a subgroup $$H$$ of $$G$$ that is local powering-invariant and such that $$Z(G) \le H$$ where $$Z(G)$$ is the center of $$G$$. A subgroup $$K$$ of $$G$$ containing $$H$$. A prime number $$p$$ such that $$K$$ is $$p$$-powered. An element $$h \in H$$.

To prove: There exists a unique element $$x \in H$$ such that $$x^p = h$$.

Proof: