This is a article which Antonio Beltran. I'm reading lemma 2.2.c). I see that:
"Lemma 2.2. Suppose that A is a finite group acting coprimely on a finite group G, and let $C = C_G(A)$. Then, for every prime p,
(b) $ν_p(C)$ divides $ν_p^A (G)$ and $ν_p^A (G)$ divides $ν_p(G)$.
(c) if N is an A-invariant normal subgroup of G, then $ν_p^A (N)$ and $ν_p^A (G/N)$ divide $ν_p^A (G)$. "
In the proof c) below. I don't understand why $C=(C\cap N)N_C(P)$ deduces $C=(C\cap N)N_C(P\cap N)$
https://www.researchgate.net/publication/291552840_Invariant_Sylow_subgroups_and_solvability_of_finite_groups[![enter image description here][2]][2]
Thank you very much.
Since $P \in Syl_p^A(G)$, and $N \unlhd G$ is $A$-invariant, we have $P \cap N$ is an $A$-invariant Sylow $p$-subgroup of $N$. The Frattini argument assures that $G=N_G(P \cap N)N$ and both $N_G(N \cap P)$ and $N$ are $A$-invariant. It follows (and this is 8.2.11 in Kurzweil-Stellmacher's The Theory of Finite Groups, an introduction, or Problem 3E.2 in Isaacs' Finite Group Theory) that $C=C_G(A)=C_{N_G(P \cap N)}C_N(A)=N_C(P \cap N)(N \cap C).$ Hope the proof is clear now, and I must admit Beltrán writes it all down somewhat confusingly.