Intermediately characteristic not implies isomorph-containing in abelian group

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 particular example::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 $$H$$.