A class of random-site mean-field Potts models is introduced and solved exactly. The bifurcation properties of the resulting mean-field equations are analysed in detail. Particular emphasis is put on the relation between the solutions and the underlying symmetries of the model. It turns out that, in contrast to the Ising case, the introduction of randomness in the Mattis-Potts model can change the order of the transition. For q<or=6 the transition becomes second order.