Verbal upper-hook verbal equals verbal in abelian group

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


 * 1) $$H$$ is a fact about::verbal subgroup of $$G$$ (which is the same as being a power subgroup, see verbal subgroup equals power subgroup in abelian group).
 * 2) There exists an abelian group $$K$$ containing $$G$$ such that both $$H$$ and $$G$$ are verbal subgroups of $$K$$.