Group property-conditionally normal-potentially relatively characteristic 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 normal-potentially relatively characteristic subgroup of $$G$$ conditional to $$\alpha$$ if there exists a group $$K$$ containing $$G$$ and satisfying $$\alpha$$ such that $$G$$ is a normal subgroup of $$G$$ and every automorphism of $$G$$ that extends to an automorphism of $$K$$ also restricts to an automorphism of $$H$$.