Subgroup of abelian group not implies abelian-potentially characteristic

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) uses::Subgroup of abelian group not implies abelian-extensible automorphism-invariant
 * 2) uses::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.