Fixers and derangements of finite permutation groups

Let $G⩽\mathrm{S}\mathrm{y}\mathrm{m}(\mathrm{\Omega })$ be a finite transitive permutation group with point stabiliser $H$. We say that a subgroup $K$ of $G$ is a fixer if every element of $K$ has fixed points, and we say that $K$ is large if $\mathrm{\mid }K\mathrm{\mid }⩾\mathrm{\mid }H\mathrm{\mid }$. There is a special interest in studying large fixers due to connections with Erdős-Ko-Rado type problems. In this paper, we classify up to conjugacy the large fixers of the almost simple primitive groups with socle ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(q)$, and we use this result to verify a special case of a conjecture of Spiga on permutation characters. We also present some results on large fixers of almost simple primitive groups with socle an alternating or sporadic group.