Verbal upper-hook verbal equals verbal in abelian group

This article gives a proof/explanation of the equivalence of multiple definitions for the term verbal subgroup of abelian group
View a complete list of pages giving proofs of equivalence of definitions

Statement

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

1. $H$ is a 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 $K$.