Geometric mean Sylow numbers of nonsolvable groups

Let G be a finite group. We write gsn(G) for the geometric mean Sylow number of G. The Fitting subgroup of Gis denoted by F(G). It was recently proved that if gsn(G)<3 root 300, then G is solvable. In this paper, we extend the study to nonsolvable groups. We prove that if G is a finite nonsolvable group, then the following holds: (a) if gsn(G)<5 root 55200 and gsn(G)not equal 4 root 2400, then G/F(G)similar to=A5; (b) if gsn(G) =4 root 2400, then G/N similar to=A5, where N is the largest normal solvable subgroup of G.