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 fc(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)).