Let G be a finite group. We write hsn(G) for the harmonic mean Sylow number of G. The Fitting subgroup of G is denoted by F(G). It is known that if \(hsn(G)<\frac{45}{7 }\) , 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 \(hsn(G)<\frac{360}{47}\) and \(hsn(G)\ne \frac{480}{71}\) , then \( G/F(G)\cong A_{5}\) ;
(b) if \(hsn(G)=\frac{480}{71}\) , then \(G/N\cong A_{5}\) , where N is the largest normal solvable subgroup of G.
