Perfect folk theorems. Does public randomization matter?

I consider two-player games, where player 1 can use only pure strategies, and player 2 can use mixed strategies. I indicate a class of such games with the property that under public randomization both the discounted and the undiscounted finitely repeated perfect folk theorems do hold, but the discounted theorem does not without public randomization. Further, I show that the class contains games such that without public randomization the undiscounted theorem does not hold, as well as games such that without public randomization the undiscounted theorem does hold.