On primitive permutation groups with a stabilizer of two points that is normal in the stabilizer of one of them: case when the socle is a power of sporadic simple group

Assume that $G$ is a primitive permutation group on a finite set $X$, $x\in X$, $y\in X\setminus\{x\}$ and $G_{x,y}\trianglelefteq G_x$. P. Cameron has raised the question about realization of an equality $G_{x,y}=1$ in this case. It is proved that, if (according to the O'Nan–Scott classification) the group $G$ is of type I, type III(a), or type III(c) or $G$ is of type II and $\operatorname{soc}(G)$ is not an exceptional group of Lie type, then $G_{x,y}=1$. In addition, it is proved that, if the group $G$ is of type III(b) and $\operatorname{soc}(G)$ is not a direct product of exceptional groups of Lie type, then $G_{x,y}=1$.