The following question is about finite groups.

Let $G$ be a finite group and let $H, K \leqslant G$. We say that $H$ permutes with $K$ if $HK = KH$ and in this case $HK \leqslant G$.

The Symbol $\pi (G)$ denotes the set of all primes that divides the order of $G$.

I need to find an example of a finite group $G$, preferably $G$ is solvable, such that the following properties hold:

(1) $G$ has a subnormal subgroup $K$.

(2) There exists a prime $p \in \pi (G)$ such that $K$ does not permute with any Sylow $p$-subgroup of $G$.

(3) $G$ has a set of nilpotent Hall subgroups $\{H_1, H_2, \dots, H_n\}$ such that $\pi (G) = \bigcup_{i = 1}^n \pi (H_i)$ and $(|H_i|,|H_j|) =1$ for $i \neq j$ and $i, j = 1, 2, \dots n$.

(4) $K$ permutes with $H_i$ for all $H_i \in \{H_1, H_2, \dots, H_n\}$.

Remark: Condition (2) implies that $|\pi (H_i)| \geqslant 2$ for at least one $i$.