On the pronormality of subgroups of odd indices in finite simple symplectic groups

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 element $g\in G$. In [Sib. Math. J. 2015. Vol. 56, no. 6] we proved that subgroups of odd indeces are pronormal in many finite simple group. In [Proc. Steklov Inst. Math., to appear] we proved that a group $PS{p}_{6n}(q)$ with $q\equiv \pm 3(\mathrm{m}\mathrm{o}\mathrm{d}8)$ contains a nonpronormal subgroup of odd index. In this paper we prove that if $n \not \in {2^m, 2^m(2^{2k}+1) \mid m, k \in $\mathbb{N}$ \cup {0}}$thenagroup$PSp_{2n}(q)$with$q \equiv \pm 3 \pmod 8$containsanonpronormalsubgroupofoddindex;if$n=2^m$ then any subgroup of odd index is pronormal in a group $PS{p}_{2n}(q)$, where $q \equiv \pm 3 \pmod 8$.