NE implies weakly normal

Statement
Any NE-subgroup of a group is a weakly normal subgroup.

NE-subgroup
A subgroup $$H$$ of a group $$G$$ is termed a NE-subgroup of $$G$$ if the intersection in $$G$$ of the normalizer $$N_G(H)$$ and the normal closure $$H^G$$ is $$H$$ itself.

Weakly normal subgroup
A subgroup $$H$$ of a group $$G$$ is termed a weakly normal subgroup of $$G$$ if any conjugate of $$H$$ that is contained in $$N_G(H)$$ is actually contained in $$H$$.

Proof
Given: A group $$G$$, a subgroup $$H$$ such that $$H = H^G \cap N_G(H)$$. A conjugate $$H^g$$ of $$H$$ such that $$H^g \le N_G(H)$$.

To prove: $$H^g \le H$$.

Proof: By definition of normal closure, $$H^g \le H^G$$. Thus, we get $$H^g \le H^G \cap N_G(H) = H$$, so $$H^G \le H$$.