Submaximal soluble subgroups of odd index in alternating groups

Let $\mathfrak{X}$ be a class of finite groups containing a group of even order and closed under subgroups, homomorphic images, and extensions. Then each finite group possesses a maximal $\mathfrak{X}$-subgroup of odd index and the study of the subgroups can be reduced to the study of the so-called submaximal $\mathfrak{X}$-subgroups of odd index in simple groups. We prove a theorem that deduces the description of submaximal $\mathfrak{X}$-subgroups of odd index in an alternating group from the description of maximal $\mathfrak{X}$-subgroups of odd index in the corresponding symmetric group. In consequence, we classify the submaximal soluble subgroups of odd index in alternating groups up to conjugacy.