On the Pronormality of Second Maximal Subgroups in Finite Groups with Socle $$L_{2}(q)$$

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