Subgroup of abelian group not implies abelian-potentially characteristic

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

It is possible to have an abelian group $G$ and a subgroup $H$ of $G$ such that there is no abelian group $K$ containing $G$ for which $H$ is a characteristic subgroup of $K$.

Facts used

1. Subgroup of abelian group not implies abelian-extensible automorphism-invariant
2. Abelian-potentially characteristic implies abelian-extensible automorphism-invariant

Proof

The proof follows from facts (1) and (2). Explicit examples include $G=\mathbb {Q} ,H=\mathbb {Z}$, and $G=\mathbb {Q} \oplus \mathbb {Q}$ and $H$ the first direct factor.