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

Abstract. A subgroup $H$ of a group $G$ is said to be pronormal in $G$ if $H$ and $H^g$ are conjugate in $\langle H, H^g \rangle$ for every $g \in G$. 
In this talk we will discuss a classification (in progress) of finite simple groups in which subgroups of odd indices are pronormal.