I came up with the following question regarding the classification of groups of order $p^2qr$ considered in pages 11,12 and 13 in the following attached article.
https://arxiv.org/pdf/1702.02616.pdf
Let $G$ be a group of order $p^2qr$, where $p,q,r$ are distinct primes. Let $F$ be a fitting subgroup of $G$. Then both $F$ 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 $\bar{F}$.
When $|F|=pr$, $\phi(F) = 1$ and $Aut(F) = C_{p−1} \times C_{r−1}$. Thus $G/F$ is abelian and hence $G/F \cong C_p \times C_q$. It follows that $p | (r − 1)$ and $q | (p − 1)(r − 1)$.
There are two cases to distinguish. First, suppose that the Sylow $q$-subgroup of $G/F$ acts non-trivially on the Sylow $p$-subgroup of $F$.
What is meant by "Sylow $q$-subgroup of $G/F$ acts non-trivially on the Sylow $p$-subgroup of $F$"? Where can I find a good knowledge about such an action?
Can someone please help to understand this question?
Thanks a lot in advance.