On orbits of automorphism groups

If $G$ is a finite group and if $A$ is a group of automorphisms of $G$ whose fixed point subgroup is $C_G(A)$ then every subgroup $F$ of $C_G(A)$ acts on the set of orbits of $A$ in $G$. The peculiarities of this action are used here to derive several results on the number of orbits of $A$ in an economical manner.