Group property-conditionally potentially fully invariant 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 fully invariant subgroup of $$G$$ conditional to $$\alpha$$ (or relative to $$\alpha$$) if there exists a group $$K$$ containing $$G$$ and satisfying $$\alpha$$ such that $$H$$ is a fully invariant subgroup of $$K$$.

Stronger properties

 * Weaker than::Group property-conditionally potentially verbal subgroup

Weaker properties

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