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As shown by Lewis, May, and McClure (MR0598689), the ordinary equivariant Bredon cohomology theory $H^*_G(-; M)$ extends to an $RO(G)$-graded cohomology theory precisely when the coefficient system $M$ extends to a Mackey functor.

Do non-ordinary equivariant cohomology theories extend to $RO(G)$-graded theories? If so, is this extension unique? If not, what can be said about these extensions?

Any references, examples, and counter-examples would be greatly appreciated.

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I believe an answer to this would also clarify this other question… – Gjergji Zaimi Jul 27 '11 at 7:45
I don't have time to think about this properly at the moment, but I think that the key is to consider functors $X\mapsto E_*(X/G)$ and $X\mapsto E^*(X/G)$ for a nonequivariant generalised (co)homology theory $E$. – Neil Strickland Jul 29 '11 at 10:17

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