Let $G=P \ltimes Q$ be a finite non-nilpotent solvable group, where $P \in {\rm Syl}_p(G)$ is cyclic. Also $P=C_G(P)=N_G(P)$ is a maximal subgroup of $G$ and $Q \cong \Bbb{Z}_q \times \Bbb{Z}_q$ is a minimal normal subgroup of $G$ such that $P$ acts irreducibly on $Q$ and $p \mid q+1$. Let $\Phi(P) \ntrianglelefteq G$ and let every non-normal subgroup of $G$ is cyclic. Now i wanted to know if $p \mid q-1$, then can we get the contradicting $\Phi(P) \trianglelefteq G$ or a contradiction with any assumption of question?
My Work. If $p \mid q-1$, then $p=2$. Let $|K|=2$. Thus there exists a subgroup $Q_1$ of order $q$ of $Q$ which is $K$-invariant. Since $KQ_1 \ntrianglelefteq G$, it must be cyclic. By Maschke Theorem there is a subgroup $Q_2$ of $Q$ such that $Q=Q_1 \times Q_2$ and $Q_2$ is $K$-invariant. Similarly $KQ_2$ is cyclic and so $K \leq Z(G)$.