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.