Complements of the socle in monolithic groups

As one of the first applications of the classification of finite simple groups, Aschbacher and Guralnick proved a cohomological statement that can be equivalently stated as follows: let H be a finite monolithic group, i.e. a group with a unique minimal normal subgroup N = soc H; if N is abelian, then the number of conjugacy classes of complements of N is H is strictly smaller than the order of N. One can ask whether the same result remains true without assuming that soc H is abelian; in other words the question is whether an analog of the theorem of Aschbacher and Guralnick holds in the non abelian context. This is the work which is performed in the present paper and which leads to the following ultimate result: Theorem. Let H be a finite group with a unique minimal normal subgroup N, which is not abelian. Then the number of conjugacy classes of complements of N in H is strictly smaller than the order of N.