On fixed-point-free involutions in actions of finite exceptional groups of Lie type

Let G be a nontrivial transitive permutation group on a finite set $Ω$. By a classical theorem of Jordan, G contains a derangement, which is an element with no fixed points on $Ω$. Given a prime divisor r of $|Ω|$, we say that G is r-elusive if it does not contain a derangement of order r. In a paper from 2011, Burness, Giudici and Wilson essentially reduce the classification of the r-elusive primitive groups to the case where G is an almost simple group of Lie type. The classical groups with an r-elusive socle have been determined by Burness and Giudici, and in this paper we consider the analogous problem for the exceptional groups of Lie type, focussing on the special case $r=2$. Our main theorem describes all the almost simple primitive exceptional groups with a $2$-elusive socle. In other words, we determine the pairs $(G,M)$, where G is an almost simple exceptional group of Lie type with socle T and M is a core-free maximal subgroup that intersects every conjugacy class of involutions in T. Our results are conclusive, with the exception of a finite list of undetermined cases for T = E₈(q), which depend on the existence (or otherwise) of certain almost simple maximal subgroups of G that have not yet been completely classified.