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

Statement
If $$K$$ is a WNSCDIN-subgroup of $$G$$, and $$H$$ is a  normalizer-relatively normal  conjugation-invariantly relatively normal subgroup of $$K$$ with respect to $$G$$, then $$H$$ is a  weakly closed subgroup of $$K$$ with respect to $$G$$.

WNSCDIN-subgroup
A subgroup $$H$$ of a group $$G$$ is a WNSCDIN-subgroup if whenever $$A,B \subseteq H$$ are normal subsets of $$H$$ and $$g \in G$$ is such that $$gAg^{-1} = B$$, there exists $$k \in N_G(H)$$ such that $$kAk^{-1} = B$$.

Normalizer-relatively normal subgroup
Suppose $$H \le K \le G$$. $$H$$ is normalizer-relatively normal in $$K$$ with respect to $$G$$ if $$H$$ is normal in $$N_G(K)$$. In other words, $$N_G(K) \le N_G(H)$$.

Conjugation-invariantly relatively normal subgroup
Suppose $$H \le K \le G$$. $$H$$ is conjugation-invariantly relatively normal in $$K$$ with respect to $$G$$ if $$H$$ is normal in every conjugate of $$K$$ in $$G$$ that contains $$H$$.

Weakly closed subgroup
Suppose $$H \le K \le G$$. $$H$$ is weakly closed in $$K$$ relative to $$G$$ if, for any $$x \in G$$ such that $$xHx^{-1} \le K$$, we have $$xHx^{-1} \le H$$.

Converse

 * Weakly closed implies normalizer-relatively normal
 * Weakly closed implies conjugation-invariantly relatively normal in finite group

Applications

 * Characteristic subgroup of Sylow subgroup is weakly closed iff it is normal in every Sylow subgroup containing it
 * Equivalence of definitions of weakly closed conjugacy functor

Proof
Given: Groups $$H \le K \le G$$ such that $$K$$ is a WNSCDIN-subgroup of $$G$$. $$H$$ is normal in $$N_G(K)$$, and $$H$$ is normal in $$gKg^{-1}$$ for all $$g \in G$$ such that $$H \le gKg^{-1}$$. We have $$x \in G$$ such that $$xHx^{-1} \le K$$.

To prove: $$xHx^{-1} \le H$$ (in fact, we'll prove equality).

Proof: