Equivalence of definitions of image-closed characteristic subgroup of finite abelian group

Statement
Suppose $$G$$ in a finite abelian group and $$H$$ is a subgroup of $$G$$. The following are equivalent:


 * 1) $$H$$ is an fact about::image-closed characteristic subgroup of $$G$$: for any surjective homomorphism from $$G$$, the image of $$H$$ is characteristic in the image of $$G$$.
 * 2) $$H$$ is an fact about::image-closed fully invariant subgroup of $$G$$: for any surjective homomorphism from $$G$$, the image of $$H$$ is fully invariant in the image of $$G$$.
 * 3) $$H$$ is a fact about::verbal subgroup of $$G$$.
 * 4) For every prime $$p$$, the $$p$$-Sylow subgroup of $$H$$ is an agemo subgroup of the $$p$$-Sylow subgroup of $$G$$.

Related facts

 * Equivalence of definitions of intermediately characteristic subgroup of finite abelian group
 * Characteristic equals fully invariant in odd-order abelian group