On the number of invariant Sylow subgroups under coprime action

Let A and G be finite groups such that A acts coprimely on G via automorphisms. We study the number of A-invariant Sylow p-subgroups of G, say νpA(G), for every prime p, and establish several arithmetical properties and formulae for these numbers. More precisely, we prove that if G is solvable and H is any A-invariant subgroup of G, then νpA(H) divides νpA(G).