On induction of class functions

Let G be a connected reductive group defined over a finite field F_q and let L be the Levi subgroup (defined over F_q) of a parabolic subgroup P of G. We define a linear map from class functions on L(F_q) to class functions on G(F_q). This map is independent of the choice of P. We show that for large q this map coincides with the known cohomological induction (whose definition involves a choice of P).