Let $G$ be finite group such that $Z(G)=1$ and the number of Sylow $p$-subgroups of $G$ equal to the number of Sylow $p-$subgroups of $PSL(2,r)$ for every prime $p$ ( $r$ is prime and $r^2$ not divide order $G$). Also $G$ has normal series $1\leq N\leq H\leq G$, such that $H/N$ is simple group and $r\mid |H/N|$. Suppose $H/N$ is abelian. If $r$ is not Merssene prime it is clear that $G$, $H$ and $N$ are unsolvable groups. I would like to know when $H/N$ is abelian, then whether there is any contradiction?
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