Normalizer-relatively normal subgroup

Definition
Suppose $$H \le K \le G$$ are groups. We say that $$H$$ is a normalizer-relatively normal subgroup of $$K$$ with respect to $$G$$ if the following equivalent conditions are satisfied:


 * Whenever $$K \le L \le G$$ is such that $$K$$ is normal in $$L$$, $$H$$ is also normal in $$L$$.
 * The normalizer $$N_G(K)$$ is contained in the normalizer $$N_G(H)$$.
 * $$H$$ is normal in $$N_G(K)$$.

Stronger properties

 * Weaker than::Middle-characteristic subgroup
 * Weaker than::Strongly closed subgroup
 * Weaker than::Weakly closed subgroup:

Weaker properties

 * Stronger than::Relatively normal subgroup