Weakly closed implies normalizer-relatively normal

This article gives the statement and possibly, proof, of an implication relation between two subgroup-of-subgroup properties. That is, it states that every subgroup-of-subgroup satisfying the first subgroup-of-subgroup property (i.e., weakly closed subgroup) must also satisfy the second subgroup-of-subgroup property (i.e., normalizer-relatively normal subgroup)
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Statement

Suppose $H\le K\le G$ are groups such that $H$ is a Weakly closed subgroup (?) of $K$ relative to $G$. Then, $H$ is a 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 $g\in N_G(K)$, $gH{g}^{-1}\le gK{g}^{-1}=K$. Thus, $gH{g}^{-1}\le K$, so by the condition of being weakly closed, $gH{g}^{-1}\le H$. Thus, $H$ is normal in $N_G(K)$.