This question was asked in https://math.stackexchange.com/questions/729648. Since I did not get any answer I am asking it here.

In an answer in Mathoverflow I see an answer but I could not understand it. May anybody help me for Understanding it. The question and its answer are:

On the character degrees of a finite group with special structure

In the beginning of the answer it is stated that by transfer and a theorem of Gaschutz we see that $N$ is complemented. By Gaschutz lemma (Stellmacher Page 74) we need an abelian and normal subgroup but $N$ is not abelian by assumption. could anybody explain why $N$ has a complement? Alex