On primitive permutation groups

Let $G$ be a primitive permutation group on a finite set $X$, $x\in X,$ $y\in X\setminus\{y\}$ and $G_{xy}\unlhd G_x$. It is proved that, if $G$ is of type I, type III(a), type III(c) (of the O'Nan–Scott classification) or $G$ is of type II and $\operatorname{soc}(G)$ is not an exceptional group of Lie type or a sporadic simple group, then $G_{xy}=1$. In addition, it is proved that if $G$ is of type III(b) and $\operatorname{soc}(G)$ is not a direct product of exceptional groups of Lie type or sporadic simple groups, then $G_{xy}=1$.