Almost elusive permutation groups

Let G be a nontrivial transitive permutation group on a finite set Ω. An element of G is said to be a derangement if it has no fixed points on Ω. From the orbit counting lemma, it follows that G contains a derangement, and in fact G contains a derangement of prime power order by a theorem of Fein, Kantor and Schacher. However, there are groups with no derangements of prime order; these are the so-called elusive groups and they have been widely studied in recent years. Extending this notion, we say that G is almost elusive if it contains a unique conjugacy class of derangements of prime order. In this paper we first prove that every quasiprimitive almost elusive group is either almost simple or 2-transitive of affine type. We then classify all the almost elusive groups that are almost simple and primitive with socle an alternating group, a sporadic group, or a rank one group of Lie type.