A classification of maximal subgroups of odd index in finite simple groups was given by Liebeck and Saxl and, independently, by Kantor in 1980s. In the cases of alternating groups or classical groups of Lie type over fields of odd characteristics, the classification was not complete.The classification was completed by the author. In the cases of finite simple classical groups of Lie type we used results obtained in Kleidman's PhD thesis. However there is a number of flaws in this PhD thesis. The flaws were corrected by Bray, Holt, and Roney-Dougal in 2013. In this note we provide a revision of the classification of maximal subgroups of odd index in finite simple classical groups over fields of odd characteristics.

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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.