Let G be a finite group with two transitive permutation representations on the sets Ω 1 and Ω 2 , respectively. We are concerned with the case that the set of fixed-point-free elements of G on Ω 1 coincides with the set of fixed-point-free elements of G on Ω 2 . We prove that if G has nilpotency class 2 then the permutation character π 1 of G on Ω 1 equals the permutation character π 2 of G on Ω 2 . Furthermore, for these groups we prove that the stabilizer of a point in Ω 1 is conjugate, under an automorphism of G, to the stabilizer of a point of Ω 2 . In Section 3 we present the following conjecture: Let G act primitively on Ω 1 and on Ω 2 and assume that the set of fixed-point-free elements of G on Ω 1 coincides with the set of fixed-point-free elements of G on Ω 2 . Then the permutation character π 1 of G on Ω 1 and the permutation character π 2 of G on Ω 2 are comparable, i.e., if π 1 ≠ π 2 then either π 1 − π 2 or π 2 − π 1 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.