Permutation characters and fixed-point-free elements in permutation groups

Let G be a finite group with two transitive permutation representations on the sets Ω1 and Ω2, respectively. We are concerned with the case that the set of fixed-point-free elements of G on Ω1 coincides with the set of fixed-point-free elements of G on Ω2. We prove that if G has nilpotency class 2 then the permutation character π1 of G on Ω1 equals the permutation character π2 of G on Ω2. Furthermore, for these groups we prove that the stabilizer of a point in Ω1 is conjugate, under an automorphism of G, to the stabilizer of a point of Ω2. In Section 3 we present the following conjecture: Let G act primitively on Ω1 and on Ω2 and assume that the set of fixed-point-free elements of G on Ω1 coincides with the set of fixed-point-free elements of G on Ω2. Then the permutation character π1 of G on Ω1 and the permutation character π2 of G on Ω2 are comparable, i.e., if π1≠π2 then either π1−π2 or π2−π1 is a character. We show that if the conjecture is false, then a minimal counterexample must be an almost simple group. Further results concerning other classes of groups are presented.