Weakly closed implies 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., relatively normal subgroup)
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Statement

Suppose $H\le K\le G$ are such that $H$ is a Weakly closed subgroup (?) of $K$ relative to $G$. Then, $H$ is a Normal subgroup (?) of $K$. In other words, $H$ is a Relatively normal subgroup (?) in $K$ with respect to $G$.