Characteristic subgroup of abelian group implies intermediately powering-invariant

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., characteristic subgroup of abelian group) must also satisfy the second subgroup property (i.e., intermediately powering-invariant subgroup)
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Statement

Suppose $G$ is an abelian group and $H,K$ are subgroups of $G$ with $H\le K\le G$. Suppose that $H$ is a characteristic subgroup of $G$. Then, $H$ is also a powering-invariant subgroup of $K$.

Related facts

• Characteristic subgroup of abelian group implies powering-invariant