Intermediately characteristic not implies isomorph-containing in abelian group

This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties, when the big group is a abelian group. That is, it states that in a abelian group, every subgroup satisfying the first subgroup property (i.e., intermediately characteristic subgroup) need not satisfy the second subgroup property (i.e., isomorph-containing subgroup)
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Definition

It is possible to have an abelian group $G$ and an intermediately characteristic subgroup $H$ of $G$ that is not an isomorph-containing subgroup of $G$.

Related facts

• Equivalence of definitions of intermediately characteristic subgroup of finite abelian group: This says that in a finite abelian group, any intermediately characteristic subgroup is an isomorph-containing subgroup.
• Characteristic equals fully invariant in odd-order abelian group
• Equivalence of definitions of image-closed characteristic subgroup of finite abelian group

Proof

Let $G$ be the group of integers under addition, and $H$ be the subgroup of even integers. Then, $H$ is a maximal subgroup of $G$, and is characteristic in $G$ (because any automorphism sends even integers to even integers). Hence, $H$ is intermediately characteristic in $G$. On the other hand, $H$ is not an isomorph-containing subgroup of $G$. For instance, $H$ is isomorphic to $G$ itself, which is not contained in $H$.