Short answer: No. Cohomology with coefficients in a ZG–module (i.e. Borel-equivariant cohomology with local coefficients pulled back from BG) is not a special case of “genuine,” RO(G)-graded equivariant cohomology. It is a different flavor of equivariant theory. It is representable, but not by an RO(G)-graded Eilenberg–MacLane G-spectrum of the sort used in May’s book; rather it is represented either in the naive (integer-graded) equivariant stable category or, more naturally, in the parametrized homotopy category over BG.

Why they are different

• For a G-space X and a ZG–module M, Brown’s H^_G(X; M) is the ordinary cohomology of the Borel construction with local coefficients determined by M: H^_G(X; M) ≅ H^*(X×_G EG; 𝓜), where 𝓜 is the local system on X×_G EG induced from the local system on BG corresponding to the G–module M.

• The “representation-graded” theory in the sense of May–Lewis–McClure–Steinberger (genuine G-theory) is represented by genuine G-spectra and is RO(G)-graded. Ordinary equivariant cohomology with coefficients in a Mackey functor Ȧ is Bredon cohomology and is represented by the equivariant Eilenberg–MacLane spectrum HȦ.

These two theories do not agree in general. A simple diagnostic:

• Take X = . Then • Borel: H^n_G(; M) ≅ H^n(BG; 𝓜) ≅ H^n(G; M) (group cohomology), often nonzero for n>0. • Genuine/Bredon: H^n_G(*; HȦ) = 0 for n>0 (a G–CW structure on * has only one 0–cell G/G). So already on a point the two theories disagree unless n=0.

What is the right relationship?

• Representability. The functor X ↦ H^n_G(X; M) is representable, but not by a genuine RO(G)-graded Eilenberg–MacLane G-spectrum. It is represented in the naive G-stable category by E_M := F(EG_+, HM), where HM is the (nonequivariant) Eilenberg–MacLane spectrum of M, endowed with the given G–action on π_0. Concretely, H^n_G(X; M) ≅ [Σ^∞ X_+, Σ^n E_M]^G in the naive equivariant stable homotopy category.

  If you insist on a genuine G-spectrum, apply the right adjoint u_* from naive to genuine G-spectra (change of universe). Then u_E_M represents the same integer-graded theory on based G–CW complexes: [Σ^∞ X_+, Σ^n u_E_M]G ≅ [Σ^∞ X+, Σ^n E_M]^G. However, the nontrivial RO(G)-degrees do not give the kind of suspension isomorphisms characteristic of genuine RO(G)-graded theories; effectively the grading collapses to the integer grading.

• Parametrized representability. Alternatively—and conceptually cleanest—cohomology with local coefficients is represented by a bundle of Eilenberg–MacLane spectra over BG in the sense of May–Sigurdsson. The ZG–module M gives a local system over BG; pulling this back along X×_G EG → BG yields a parametrized EM-spectrum whose (nonequivariant) cohomology is precisely H^*_G(X; M).

• Natural comparison from genuine theories. For any genuine G-spectrum E there is a canonical “Borelization” natural transformation E_G^(X) = [Σ^∞ X_+, Σ^ E]_G → E^*(X×G EG) obtained by smashing with EG+ and then forgetting equivariance. For E = HȦ this gives a natural map from Bredon cohomology to Borel cohomology; it is seldom an isomorphism.

About your Z/2 example

Your isomorphism H^m_{Z/2}(X; I^{⊗ m}) ≅ H^{mT}{Z/2}(X+) for free actions comes from two special features:

• free G-action (so everything is controlled by the classifying map X/G → BG), and
• the fact that the augmentation ideal I corresponds to the sign line T (a 1-dimensional real G-representation), giving a twisted Thom isomorphism.

This kind of identification relies on having a genuine 1-dimensional representation whose orientation local system is the given module. Most ZG–modules do not arise as such orientation systems, so one should not expect a general “twisted Thom” identification with RO(G)-graded cohomology.

Takeaways

• Borel-equivariant cohomology with coefficients in a ZG–module is not a special case of genuine RO(G)-graded cohomology with a Mackey functor. They agree only in very special cases.
• It is representable: • by a naive G-spectrum F(EG_+, HM) (or, if desired, by the cofree genuine G-spectrum u_*F(EG_+, HM)), and • more naturally by a parametrized Eilenberg–MacLane spectrum over BG.
• There is a canonical comparison from genuine RO(G)-graded theories to Borel cohomology, but it is not an isomorphism in general.

References for the representability statements:

• K. Brown, Cohomology of Groups, for the Borel/coefficient point of view.
• J. P. May, Equivariant Homotopy and Cohomology Theory, esp. the discussion of Borel cohomology and naive vs genuine G-spectra (see the change-of-universe adjunctions).
• J. P. May and J. Sigurdsson, Parametrized Homotopy Theory, for representability of cohomology with local coefficients by parametrized Eilenberg–MacLane spectra (see e.g. Example 23.3.10).