Isomorph-normal characteristic of WNSCDIN implies weakly closed

Statement

Suppose $H\leq K\leq G$ are groups. Suppose, further, that $H$ is an Isomorph-normal characteristic subgroup (?) of $K$ and $K$ is a WNSCDIN-subgroup (?) of $G$. Then, $H$ is a Weakly closed subgroup (?) of $K$ relative to $G$.

Facts used

1. Characteristic of normal implies normal
2. WNSCDIN implies every normalizer-relatively normal conjugation-invariantly relatively normal subgroup is weakly closed

Proof

Given: Groups $H\leq K\leq G$, 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 $H\leq gKg^{-1}$ for some $g\in G$. Then, $g^{-1}Hg\leq K$. 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.