Local powering-invariance is quotient-transitive in nilpotent group

Statement
Suppose $$G$$ is a nilpotent group with subgroups $$H \le K$$ satisfying the following:


 * $$H$$ is a normal subgroup of $$G$$.
 * $$H$$ is a local powering-invariant subgroup of $$G$$, i.e., for $$h \in H$$ and $$n \in \mathbb{N}$$ such that the equation $$x^n = h$$ has a unique solution for $$x$$ in $$G$$, we must have $$x \in H$$.
 * $$K/H$$ is a local powering-invariant subgroup of the quotient group $$G/H$$.

Then, $$K$$ is a local powering-invariant subgroup of $$G$$.

Applications

 * Upper central series members are local powering-invariant in nilpotent group

Opposite facts

 * Second center not is local powering-invariant in solvable group

Facts used

 * 1) uses::Local powering-invariant and normal iff quotient-local powering-invariant in nilpotent group
 * 2) uses::Local powering-invariant over quotient-local powering-invariant implies local powering-invariant

Proof
The proof follows directly by combining Facts (1) and (2).