Paranormal implies weakly closed in intermediate nilpotent

Statement with symbols
Suppose $$H \le K \le G$$ are groups such that:


 * $$H$$ is a fact about::paranormal subgroup of $$G$$.
 * $$K$$ is a fact about::nilpotent group.

Then, $$H$$ is a fact about::weakly closed subgroup of $$K$$.

Related facts

 * Paranormal implies weakly normal
 * Paranormal implies intermediately subnormal-to-normal

Corollaries

 * Pronormal implies weakly closed in intermediate nilpotent

Definitions used
For these definitions, $$H^g = g^{-1}Hg$$ denotes the conjugate subgroup by $$g \in G$$. (This is the right-action convention; however, adopting a left-action convention does not alter any of the proof details).

Paranormal subgroup
A subgroup $$H$$ of a group $$G$$ is termed paranormal in $$G$$ if for any $$g \in G$$, $$H$$ is a contranormal subgroup of $$\langle H, H^g \rangle$$; in other words, the normal closure of $$H$$ in $$\langle H, H^g \rangle$$ is the whole of $$\langle H, H^g \rangle$$.

Weakly closed subgroup
Suppose $$H \le K \le G$$ are groups. We say $$H$$ is weakly closed in $$K$$ with respect to $$G$$ if, for any $$g \in G$$ such that $$H^g \le K$$, we have $$H^g \le H$$.

Facts used

 * 1) uses::Paranormal implies weakly normal
 * 2) uses::Weakly normal implies weakly closed in intermediate nilpotent

Proof
The proof follows directly from facts (1) and (2).