Verbal-potentially verbal subgroup

Definition
A subgroup $$H$$ of a group $$G$$ is termed a verbal-potentially verbal subgroup if there exists a group $$K$$ containing $$G$$ such that both $$H$$ and $$G$$ are fact about::verbal subgroups of $$K$$.

In terms of the upper-hook operator
Given subgroup properties $$p$$ and $$q$$, the upper-hook of $$p$$ and $$q$$ is defined as the following subgroup property $$r$$: a subgroup $$H$$ of a group $$G$$ satisfies the property $$r$$ if there exists a group $$K$$ containing $$G$$ such that $$H$$ satisfies property $$p$$ in $$K$$ and $$G$$ satisfies property $$q$$ in $$K$$.

Facts

 * Verbal upper-hook verbal equals verbal in abelian group