Finite non-solvable groups with few 2-parts of co-degrees of irreducible characters

For a character of a finite group G, the number xc(1) = [G:ker ] x (1) is called the co-degree of x . Let Sol(G) denote the solvable radical of G. In this paper, we show that if G is a finite non-solvable group with f c(1)2 :x ∈ Irr(G)g = f1; 2mg for some positive integer m, then G=Sol(G) has a normal subgroup M=Sol(G) such that M=Sol(G) = PSL2(2n) for some integer n 2, [G : M] is odd and G=Sol(G) . Aut(PSL2(2n)).