In 1972, Hering classified the finite doubly transitive permutation groups $(G,X)$ ($G$ acting faithfully on $X$) in which $G_x$, with $x \in X$, contains a normal subgroup $N_x$ of even order which acts semiregularly on $X \setminus \{x\}$ (without CFSG).
Is such a classification (without CFSG) known when $\vert N_x \vert$ is odd ($N_x$ still acting semiregularly on $X \setminus \{ x\}$) ?
