Powering-invariance is transitive

Statement
Suppose $$H \le K \le G$$ are groups such that $$H$$ is a powering-invariant subgroup of $$K$$ and $$K$$ is a powering-invariant subgroup of $$G$$. Then, $$H$$ is a powering-invariant subgroup of $$G$$.

Related facts

 * Divisibility-invariance is transitive
 * Powering-invariance is not quotient-transitive