Let $G$ be a solvable, non-nilpotent group of order $p^2qr$, where $p,q,r$ are distinct primes, and let $F$ be a Fitting subgroup of $G$. Then $F$ and $G/F$ are both non-trivial and $G/F$ acts faithfully on $\bar{F}:=F/\phi(F)$ so that no non-trivial normal subgroup of $G/F$ stabilizes a series through $\bar{F}$.
Can someone please help me to understand how to write the group $G$ using notations , when $|F|=pr$? Is it correct if I say $G \cong (C_p \times C_r) \rtimes (C_p \times C_q)$ or $G \cong C_{pr} \rtimes (C_p \times C_q)$ ?
Note: When $|F|=pr$, $\phi(F)=1$ and $Aut(F) = C_{p-1} \times C_{q-1}$. Thus $G/F$ is abelian and $G/F \cong (C_p \times C_q)$.
Thanks a lot in advance.
(The description about group $G$ is related to the information in page 12, last bullet point in the following pdf).