Local powering-invariance is transitive

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

Proof
The proof involves standard element-chasing.