Powering-invariance does not satisfy intermediate subgroup condition

This article gives the statement, and possibly proof, of a subgroup property (i.e., powering-invariant subgroup) not satisfying a subgroup metaproperty (i.e., intermediate subgroup condition).
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Statement

It is possible to have groups $H\leq K\leq G$ 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 $K$.