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Hall property Subgroup of odd index Classical group Group of Lie type In this paper we find the number of classes of conjugate π -Hall subgroups in all finite almost simple groups. We also complete the classification of π -Hall subgroups in finite simple groups and correct some mistakes from our previous paper.

A subgroup H of a group G is said to be pronormal in G if H and H g are conjugate in H, H g for every g ∈ G. In [Sib. Math. J. 2012. Vol. 53, no. 3], the following conjecture was formulated by Evgeny P. Vdovin and the third author.Conjecture. All subgroups of odd indices are pronormal in all finite simple groups.The conjecture was verified by authors for many families of finite simple groups in [Sib. Math. J. 2015. Vol. 56, no. 6]. Namely, the following theorem was proved.Theorem. All subgroups of odd indices are pronormal in the following finite simple groups: A n , where n ≥ 5; sporadic groups; groups of Lie type over fields of characteristic 2; L 2 n (q); U 2 n (q); S 2n (q), where q ≡ ±3 (mod 8); O n (q); exeptional groups of Lie type not isomorphic to E 6 (q) or 2 E 6 (q).In [Proc. Steklov Inst. Math., to appear, Theorem 1] authors proved that, if a group G has a normal abelian subgroup V and a subgroup H such that G = HV , then H is pronormal in G if and only if U = N U (H)[H, U ] for any H-invariant subgroup U of the group V . Using this fact, in [Proc. Steklov Inst. Math., to appear, Theorem 2] it was proved that Conjecture fails. Precisely, a finite simple symplectic group P Sp 6n (q) with q ≡ ±3 (mod 8) contains a nonpronormal subgroup of odd index. In view of the above results, the following problem naturally arises.Problem. Classify finite simple groups in which all subgroups of odd indices are pronormal.Using [Proc. Steklov Inst. Math., to appear, Theorem 2] we prove the following theorem. Theorem 1. Let G = P Sp 2n (q), where q ≡ ±3 (mod 8) and n ∈ {2 m , 2 m (2 2k + 1) | m, k ∈ N ∪ {0}}. Then G contains a nonpronormal subgroup of odd index.The main result of this paper is the following theorem.Theorem 2. Let G = P Sp 2 n (q), where n ≥ 2 and q ≡ ±3 (mod 8). Then any subgroup of odd index of G is pronormal in G.