Complements of the socle in almost simple groups

Assume that a finite group H has a unique minimal normal subgroup, say N, and that N has a complement in H. We want to bound the number of conjugacy classes of complements of N in H; in particular we are looking for a bound which depends on the order of N. When N=socH is abelian, the conjugacy classes of complements of N in H are in bijective correspondence with the elements of the first cohomology group H1 (H/N, N). Using the classification of finite simple groups, Aschbacher and Guralnick proved that |H1 (H/N, N)|<|N|; therefore, when socH=N is abelian, there are at most |N| conjugacy classes of complements of N in H. We conjecture that the same result holds also in the case when N=socH is nonabelian. In this paper we deal with this conjecture in the case of finite almost simple groups. Let G be a finite simple group; we may identify G with Inn (G). We will prove the following THEOREM. Let G be a finite non-abelian simple group and assume that the subgroup H of Aut (G) contains G. Then the number of conjugacy classes of complements of G in H is less than |G|.