This is a article which Gabriel Navarro wrote. I'm reading lemma 2.1. I see that
"By standard arguments, recall that in any coprime action, if $q$ is a prime, then every $A$-invariant $q$-subgroup of $G$ is contained in an $A$-invariant Sylow $q$-subgroup of $G$. Furthermore, any two $A$-invariant Sylow $q$-subgroups are $C$-conjugate and if $P$ $\in$ $Syl_q(G)$ is $A$-invariant, then $P \cap C \in Syl_q(C)$."
I understand that by the Sylow theorems, every $q$-subgroup of $G$ is contained in a Sylow $q$-subgroup of $G$, and that any two Sylow $q$-subgroups are conjugate
But I have no experience with coprime actions, and I don't understand the $A$-invariant counterparts.
I would also appreciate any documentation about coprime actions and their basic properties.
Thank you.
https://www.ams.org/journals/proc/2003-131-10/S0002-9939-03-06884-9/S0002-9939-03-06884-9.pdf
As a good introduction, read chapter $3E$ of M.I. Isaacs' Finite Group Theory. This will give you information about Sylow theory under co-prime actions. By the way, from Navarro's beautiful paper it follows that in solvable groups $G$ for every $H \leq G$ and every prime $p$ dividing $|G|$, $n_p(H) \mid n_p(G)$ (where $n_p(\cdot)$ denotes the number of Sylow $p$-subgroups).
Also, the following posts are useful to read:
$A$-invariant Sylow $p$-group
Number of Sylow subgroups in $p$-solvable groups - paper of G. Navarro
and bearing the same title but different entry:
Number of Sylow subgroups in $p$-solvable groups - paper of G. Navarro
The article Number of Sylow subgroups in $p$-solvable groups - paper of G. Navarro