Conjugation-invariantly embedded normal subgroup

Definition
Suppose $$H \le K \le G$$ are groups. We say that $$H$$ is a conjugation-invariantly embedded normal subgroup of $$K$$ with respect to $$G$$ if $$H$$ is a normal subgroup of $$K$$, and further, if $$L$$ is a conjugate subgroup to $$K$$ containing $$H$$, then there exists an isomorphism $$\sigma:K \to L$$ such that $$\sigma(H) = H$$.

In other words, $$H \le K \le G$$ is both conjugation-invariantly embedded and relatively normal.

Stronger properties

 * Weaker than::Strongly closed subgroup
 * Weaker than::Weakly closed subgroup

Weaker properties

 * Stronger than::Conjugation-invariantly relatively normal subgroup
 * Stronger than::Relatively normal subgroup