Quotient-local powering-invariance is quotient-transitive

Statement
Suppose $$G$$ is a group and $$H,K$$ are subgroups of $$G$$ with $$H \le K$$. Suppose $$H$$ is a quotient-local powering-invariant subgroup of $$G$$ and $$K/H$$ is a quotient-local powering-invariant subgroup of the quotient group $$G/H$$. Then, $$K$$ is a quotient-local powering-invariant subgroup of $$G$$.

Similar facts

 * Local powering-invariant over quotient-local powering-invariant implies local powering-invariant

Applications

 * Upper central series members are quotient-local powering-invariant in nilpotent group
 * Local powering-invariance is quotient-transitive in nilpotent group

Proof
The proof is straightforward, using element chasing.