Weakly closed implies normalizer-relatively normal

Statement
Suppose $$H \le K \le G$$ are groups such that $$H$$ is a fact about::weakly closed subgroup of $$K$$ relative to $$G$$. Then, $$H$$ is a fact about::normalizer-relatively normal subgroup of $$K$$ relative to $$G$$: in other words, $$H$$ is a normal subgroup of $$N_G(K)$$.

Related facts

 * WNSCDIN implies every normalizer-relatively normal conjugation-invariantly relatively normal subgroup is weakly closed

Proof
Given: $$H \le K \le G$$ such that $$H$$ is weakly closed in $$K$$ relative to $$G$$.

To prove: $$H$$ is normal in $$N_G(K)$$.

Proof: For $$g \in N_G(K)$$, $$gHg^{-1} \le gKg^{-1} = K$$. Thus, $$gHg^{-1} \le K$$, so by the condition of being weakly closed, $$gHg^{-1} \le H$$. Thus, $$H$$ is normal in $$N_G(K)$$.