Motivated by a conjecture of Klapper [Finite Fields, Coding Theory, and Advances in Communications and Computing, Marcel Dekker, New York, 1993], we study the distribution of elements ζ of a finite field F of qn elements under the action of the transformations ? -· (aζ + b)/(cζ + d) for matrices ( acbd) ε PGL2(q). We slightly improve a result of Niederreiter and Winterhof [Finite Fields Appl, 9 (2003), pp. 458-471] towards this conjecture. On the other hand, we also show that the original conjecture is false as stated.