Isomorph-normal characteristic of WNSCDIN implies weakly closed

Statement
Suppose $$H \le K \le G$$ are groups. Suppose, further, that $$H$$ is an fact about::isomorph-normal characteristic subgroup of $$K$$ and $$K$$ is a fact about::WNSCDIN-subgroup of $$G$$. Then, $$H$$ is a fact about::weakly closed subgroup of $$K$$ relative to $$G$$.

Facts used

 * 1) uses::Characteristic of normal implies normal
 * 2) uses::WNSCDIN implies every normalizer-relatively normal conjugation-invariantly relatively normal subgroup is weakly closed

Proof
Given: Groups $$H \le K \le G$$, such that $$H$$ is characteristic in $$K$$, every subgroup of $$K$$ isomorphic to $$H$$ is normal in $$K$$, and $$K$$ is a WNSCDIN-subgroup of $$G$$.

To prove: $$H$$ is weakly closed in $$G$$.

Proof:


 * 1) $$H$$ is normal in $$N_G(K)$$: Since $$H$$ is characteristic in $$K$$ and $$K$$ is normal in $$N_G(K)$$, fact (1) yields that $$H$$ is normal in $$N_G(K)$$.
 * 2) $$H$$ is normal in every conjugate of $$K$$ containing it: Suppose $$H \le gKg^{-1}$$ for some $$g \in G$$. Then, $$g^{-1}Hg \le K$$. Clearly, $$g^{-1}Hg$$ is isomorphic to $$H$$. So, by the assumption, $$g^{-1}Hg$$ is normal in $$K$$. Conjugating back, we get that $$H$$ is normal in $$gKg^{-1}$$.
 * 3) $$H$$ is weakly closed in $$K$$ with respect to $$G$$: This follows from fact (2), using the previous two steps.