On the Pronormality of Second Maximal Subgroups in Finite Groups with Socle $L_2(q)$

According to P. Hall, a subgroup $H$ of a finite group $G$ is called pronormal in $G$ if, for any element $g$ of $G$, the subgroups $H$ and $H^g$ are conjugate in $\langle H,H^g\rangle$. The simplest examples of pronormal subgroups of finite groups are normal subgroups, maximal subgroups, and Sylow subgroups. Pronormal subgroups of finite groups were studied by a number of authors. For example, Legovini (1981) studied finite groups in which every subgroup is subnormal or pronormal. Later, Li and Zhang (2013) described the structure of a finite group $G$ in which, for a second maximal subgroup $H$, its index in $\langle H,H^g\rangle$ does not contain squares for any $g$ from $G$. A number of papers by Kondrat'ev, Maslova, Revin, and Vdovin (2012–2019) are devoted to studying the pronormality of subgroups in a finite simple nonabelian group and, in particular, the existence of a nonpronormal subgroup of odd index in a finite simple nonabelian group. In {The Kourovka Notebook}, the author formulated Question 19.109 on the equivalence in a finite simple nonabelian group of the condition of pronormality of its second maximal subgroups and the condition of Hallness of its maximal subgroups. Tyutyanov gave a counterexample $L_2(2^{11})$ to this question. In the present paper, we provide necessary and sufficient conditions for the pronormality of second maximal subgroups in the group $L_2(q)$. In addition, for $q\le 11$, we find the finite almost simple groups with socle $L_2(q)$ in which all second maximal subgroups are pronormal.