Conjugation-invariantly embedded subgroup

Definition
Suppose $$H \le K \le G$$ are subgroups. We say that $$H$$ is conjugation-invariantly embedded in $$K$$ with respect to $$G$$ if, for any conjugate $$L$$ of $$K$$ in $$G$$ that contains H, there is an isomorphism $$\sigma: K \to L$$ such that $$\sigma(H) = H$$.

Stronger properties

 * Weaker than::Weakly closed subgroup:
 * Weaker than::Conjugation-invariantly embedded normal subgroup