Group property-conditionally potentially verbal subgroup

Definition
Suppose $$\alpha$$ is a group property, $$G$$ is a group satisfying $$\alpha$$, and $$H$$ is a subgroup of $$G$$. We say that $$H$$ is a potentially verbal subgroup conditional to $$\alpha$$ if there exists a group $$K$$ containing $$G$$ and satisfying $$\alpha$$ such that $$H$$ is a verbal subgroup of $$K$$.

Weaker properties

 * Stronger than::Group property-conditionally potentially fully invariant subgroup
 * Stronger than::Group property-conditionally potentially characteristic subgroup