Kronecker classes and cliques in derangement graphs

Given a permutation group $G$, the derangement graph of $G$ is defined with vertex set $G$, where two elements $x$ and $y$ are adjacent if and only if $x{y}^{-1}$ is a derangement. We establish that, if $G$ is transitive with degree exceeding 30, then the derangement graph of $G$ contains a complete subgraph with four vertices. As a consequence, if $G$ is a normal subgroup of $A$ such that $\mathrm{\mid }A:G\mathrm{\mid }=3$, and if $U$ is a subgroup of $G$ satisfying $G={\bigcup }_{a\in A}{U}^a$, then $\mathrm{\mid }G:U\mathrm{\mid }\le 10$. This result provides support for a conjecture by Neumann and Praeger concerning Kronecker classes.