Characteristic subgroup of abelian group implies intermediately powering-invariant

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