In their paper, "The Folk Theorem with Imperfect Public Information", several versions of Folk Theorem are shown. This is a memo for them. Before mentioning actual theorem, let's briefly check their contribution in the literature.

An important hypothesis of the standard Folk Theorem is that the players can observe one another's actions in each repetition, so that deviations from equilibrium strategies are detectable. In contrast, this paper considers games in which players observe only a public outcome that is a stochastic function of the actions played. Thus these are games of moral hazard. The major task of the paper is to provide conditions sufficient for the Folk Theorem to extend to such games. The most important hypotheses concern the way the probability distribution over public outcomes depends on the player's actions.

To see the Folk Theorem with perfect information, you should check Fudenberg and Maskin (Econometrica 1986) "The Folk Theorem for Repeated Games with Discounting and Incomplete Information". (Notice that "Incomplete Information" in the title implies the cases without a public randomization device or some reputation models and do not mean imperfect monitoring cases, which is covered in FLM)

The three versions of Folk Theorem they showed are as follows:

Nash-thread 1
If all pure-action profiles are pairwise identifiable, a "Nash-threat" version of the Folk Theorem obtains: any payoff vector Pareto-dominating a Nash equilibrium of the stage game can be sustained in an equilibrium of the repeated game for discount factors near enough to 1.

Nash-thread 2
If a game has at least one (mixed-action) profile satisfying the conjunction of pairwise identifiability and individual full rank (= "pairwise full rank"), then again the Nash-threat Folk Theorem applies. Generic games possess such a profile provided that the number of possible public outcomes is no less than the total number of elements in the action sets of any two players.

Minimax-thread
To obtain the conventional "minimax-threat" Folk Theorem requires more stringent conditions. Specifically, besides the hypotheses for the Nash-threat theorem, it suffices to assume that all pure-action profiles satisfy individual full rank.

Finally, I quote their explanation about two examples of inefficient results shown by Radner-Myerson-Maskin and Green-Porter (or Abreu-Pearce-Stacchetti)

Our work makes clear that the R-M-M counterexample relies on there being relatively few possible public outcomes compared to the number of possible actions, so that the genericity result mentioned before does not apply, and that equilibria of the G-P and A-P-S variety are necessarily inefficient only because they are symmetric: if (even slightly) asymmetric equilibria are admitted, the Folk Theorem is restored.