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 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 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 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