Powering-invariance does not satisfy intermediate subgroup condition

Statement
It is possible to have groups $$H \le K \le G$$ such that $$H$$ is a powering-invariant subgroup of $$G$$ but not of $$K$$.

Proof
Take:


 * $$G$$ to be the group $$\mathbb{Q} \oplus \mathbb{Z}$$.
 * $$K$$ to be the subgroup $$\mathbb{Q} \oplus 0$$.
 * $$H$$ to be the subgroup $$\mathbb{Z} \oplus 0$$.

Then:


 * $$H$$ is powering-invariant in $$G$$: $$G$$ is not powered over any primes, so $$H$$ is by definition powering-invariant in $$G$$.
 * $$H$$ is not powering-invariant in $$K$$: $$K$$ is powered over all primes, and $$H$$ is not powered over any, so $$H$$ is not powering-invariant in $$K$$.