Relatively maximal subgroups of odd index in symmetric groups

Let $\mathfrak{X}$ be a class of finite groups which contains a group of order $2$ and is closed under subgroups, homomorphic images, and extensions. We define the concept of an $\mathfrak{X}$-admissible diagram representing a natural number $n$. Associated with each $n$ are finitely many such diagrams, and they all can be found easily. Admissible diagrams representing a number $n$ are used to uniquely parametrize conjugacy classes of maximal $\mathfrak{X}$-subgroups of odd index in the symmetric group $\mathrm{Sym}_n$, and we define the structure of such groups. As a consequence, we obtain a complete classification of submaximal $\mathfrak{X}$-subgroups of odd index in alternating groups.