A solvable group of order $mn$ with $m,n>1$ and $(m,n)=1$, where $|H|=m, |K|=n$ for $H,K\le G$.

Question

Let $G$ be a solvable group of order $mn$, with $m,n>1$ and $(m,n)=1$. Let $H,K \leq G$ be such that $|H|=m$ and $|K|=n$. Show that one of the following items is valid:
(i) The normal closure $H^G<G$ and $N_G(H)>H$;
(ii) The normal closure $K^G<G$ and $N_G(K)>K$.

I showed that $G=H^GN_G(H)$ and $G=K^GN_G(K)$ using Hall's Theorem for solvable groups (existence and conjugation of Hall subgroups). If $H^G<G$, then $N_G(H)>H$, but if $H^G=G$, I am not able to show that item (ii) follows.

Answer 1

Since $G$ is solvable, it has a normal subgroup $N$ such that $|G/N| = p$ is prime. Then $p$ must divide $m$ or $n$

If $p$ divides $n$ then $H \le N $, so $H^G \le N < G$.

Similarly, if $p$ divides $m$ then $K^G \le N < G$.