On a conjecture on the permutation characters of finite primitive groups

Let be a finite group with two primitive permutation representations on the sets and and let and be the corresponding permutation characters. We consider the case in which the set of fixed-point-free elements of on coincides with the set of fixed-point-free elements of on, that is, for every, if and only if. We have conjectured in Spiga ['Permutation characters and fixed-point-free elements in permutation groups', J. Algebra299(1) (2006), 1-7] that under this hypothesis either or one of and is a genuine character. In this paper we give evidence towards the veracity of this conjecture when the socle of is a sporadic simple group or an alternating group. In particular, the conjecture is reduced to the case of almost simple groups of Lie type.