2-hypernormalized satisfies intermediate subgroup condition

Statement
Suppose $$H$$ is a 2-hypernormalized subgroup of a group $$G$$. In other words, $$N_G(N_G(H)) = G$$. Suppose $$K$$ is a subgroup of $$G$$ containing $$H$$. Then, $$H$$ is 2-hypernormalized in $$K$$: $$N_K(N_K(H)) = K$$.

Related facts

 * Finitarily hypernormalized does not satisfy intermediate subgroup condition
 * 2-hypernormalized does not satisfy transfer condition

Facts used

 * 1) uses::Normality satisfies transfer condition

Proof
Given: $$H \le K \le G$$, $$N_G(N_G(H)) = G$$.

To prove: $$N_K(N_K(H)) = K$$.

Proof:


 * 1) $$N_K(H) = N_G(H) \cap K$$: This follows directly from the definition of normalizer.
 * 2) $$N_K(H)$$ is normal in $$K$$: By assumption, $$N_G(H)$$ is normal in $$G$$. Fact (1) yields that $$N_G(H) \cap K = N_K(H)$$ is normal in $$K$$.
 * 3) $$N_K(N_K(H)) = K$$: This follows directly from the previous step.