Equivalence of definitions of intermediately characteristic subgroup of finite abelian group

Statement
The following are equivalent for a subgroup $$H$$ of a finite abelian group $$G$$:


 * 1) $$H$$ is an fact about::intermediately characteristic subgroup of $$G$$, i.e., $$H$$ is a characteristic subgroup in every subgroup of $$G$$ containing it.
 * 2) $$H$$ is an fact about::intermediately fully invariant subgroup of $$G$$, i.e., $$H$$ is a fully invariant subgroup in every subgroup of $$G$$ containing it.
 * 3) $$H$$ is an fact about::isomorph-containing subgroup of $$G$$ -- it contains every subgroup of $$G$$ isomorphic to it. Equivalently, $$H$$ is an fact about::isomorph-free subgroup of $$G$$.
 * 4) $$H$$ is a fact about::homomorph-containing subgroup of $$G$$ -- it contains every subgroup of $$G$$ that is a homomorphic image of it.
 * 5) For every prime $$p$$, the $$p$$-Sylow subgroup of $$H$$ is an omega subgroup of the corresponding $$p$$-Sylow subgroup of $$G$$.

Related facts

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