Classification of finite solvable CN-groups

Statement
Suppose $$G$$ is a finite group that is both a uses property satisfaction of::finite solvable group and a uses property satisfaction of::CN-group  (hence also a  finite CN-group. Then, $$G$$ must be of one of these three types:


 * 1) $$G$$ is a proves property satisfaction of::finite nilpotent group.
 * 2) $$G$$ is a proves property satisfaction of::Frobenius group  where the Frobenius complement is a proves property satisfaction of::finite group in which every abelian subgroup is cyclic.
 * 3) $$G$$ is a proves property satisfaction of::3-step group.