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 $H$.