Nonsolvable groups whose irreducible character degrees have special 2-parts

Let G be a nonsolvable group and Irr(G) the set of irreducible complex characters of G. We consider the nonsolvable groups whose character degrees have special 2-parts and prove that if χ(1)2 = 1 or ∣G∣2 for every χ ∈ Irr(G), then there exists a minimal normal subgroup N of G such that N ≅ PSL(2, 2n) and G/N is an odd order group.