A characterization of simple Zassenhaus groups

Let a finite group $G$ have a $CC$-subgroup $M$ of order $m$ whose normalizer differs from $M$ and $G$, and let the order of $N_G(M)$ be odd and each coset $Mx$ of $G$, for $x\in G\setminus N_G(M)$, contain an involution. Earlier the author (R Zh Mat, 1979, 8A154) posed the question of the existence of simple groups other than $PSL(2,m)$ with the indicated properties. In this paper it is proved that $G\cong PSL(2,m)$. The result includes theorems of Feit and Ito on Zassenhaus groups.
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