Let \(G\leqslant \textrm{Sym}(\Omega )\) 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 \(|K| \geqslant |H|\) . 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 \(\textrm{PSL}_2(q)\) , 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.
