Group property-conditionally normal-potentially characteristic subgroup

Definition
Let $$H$$ be a subgroup of a group $$G$$, and $$\alpha$$ be a group property. We say that $$H$$ is normal-potentially characteristic in $$G$$ relative to $$\alpha$$ if there exists a group $$K$$ containing $$G$$ and satisfying $$\alpha$$ such that $$G$$ is a normal subgroup of $$K$$ and $$H$$ is a characteristic subgroup of $$K$$.

If we simply use the term normal-potentially characteristic subgroup, it means that $$\alpha$$ is taken to be the tautology -- the property satisfied by all groups.

Stronger properties

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

Weaker properties

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