Group property-conditionally potentially characteristic subgroup

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

Stronger properties

 * Weaker than::Group property-conditionally potentially verbal subgroup
 * Weaker than::Group property-conditionally potentially fully invariant subgroup
 * Weaker than::Group property-conditionally normal-potentially characteristic subgroup
 * Weaker than::Group property-conditionally characteristic-potentially characteristic subgroup