Equivalence classes of primes under commuting equivalence relation on finite CN-group give Hall subgroups that are nilpotent and TI

Statement
Suppose $$G$$ is a finite CN-group (i.e., a finite group that is also a CN-group). Let $$\pi$$ be the set of prime divisors of the order of $$G$$. Define a relation $$\sim$$ on $$\pi$$ as follows: for $$p,q \in \pi$$ (possibly equal, possibly distinct), $$p \sim q$$ if and only if either $$p = q$$ or there exists a $$p$$-Sylow subgroup $$P$$ of $$G$$ and a $$q$$-Sylow subgroup $$Q$$ of $$G$$ such that the following equivalent conditions hold:


 * 1) There exists a non-identity element $$x$$ of $$P$$ and a non-identity element $$y$$ of $$Q$$ such that $$x$$ and $$y$$ commute.
 * 2) Every element of $$P$$ commutes with every element of $$Q$$.

By uses::commuting of non-identity elements defines an equivalence relation between prime divisors of the order of a finite CN-group, $$\sim$$ is an equivalence relation.

The current claim is as follows: for any equivalence class under $$\omega$$, $$G$$ admits $$\omega$$-Hall subgroups that are nilpotent and TI.