  Let $G\leqslant\mathrm{Sym}(\Omega)$ 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 Erd\H{o}s-Ko-Rado type problems. In this paper, we classify up
to conjugacy the large fixers of the almost simple primitive groups with socle
$\mathrm{PSL}_2(q)$, 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.
