New examples of nonpronormal subgroups of odd index in finite simple linear and unitary groups

A subgroup $H$ of a group $G$ is said to be pronormal if, for every element $g \in G$, the subgroups $H$ and $H^g$ are conjugate in the subgroup $\langle H, H^g\rangle$. It is known that a substantial portion of finite simple groups possesses property $(*)$: every subgroup of odd index is pronormal in the group. To date, finite simple groups with property $(*)$ have been classified, except for finite simple linear and unitary groups subject to certain restrictions on their natural arithmetic parameters. In 2024, a classification was initiated for finite simple linear and unitary groups in which all subgroups of odd index are pronormal. The plan is to identify all possible sources of nonpronormal subgroups of odd index and then prove that there are no other such examples. In 2024, series of examples of nonpronormal subgroups of odd index were found in finite simple linear and unitary groups over fields of odd characteristic. In the present paper, we construct a new series of examples of nonpronormal subgroups of odd index in finite simple linear and unitary groups over a field of odd characteristic.