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This is not a helpful answer to your main question, but merely a negative answer to your "In particular..what happens.." question. But, the general idea may be helpful in figuring out what more preci …

I'd like to propose that the answer is "No, but..". The viewpoint I'd like to suggest is that thinking of "derived functors" is probably insufficient here (because we're secretly interested in homoto …

As I posted in my comment, I think Paul's suggestion does work. Here's a (sloppy) description of how I think things will work: The local systems you describe can be obtained, by passing to homotopy …

In the model structure you describe, the cofibrations should be the retracts of the free relative G-cell maps: i.e., retracts of maps obtained by attaching cells of the form $G \times S^{n} \to G \tim …

There's also a different way of writing down the $H$-space structure, that I like for its algebro-geometric flavor. (I'll talk about $\mathbb{C}P^\infty$ here, and $\mathbb{R}P^\infty$ should be anal …

I don't know if this is exactly what you're looking for (and there's a good chance you already know what I'm going to write) but let me give it a try: The realization functor of cyclic sets (not spac …