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I would like to have proper references in a paper that I'm writing down. This concerns computations of the coefficients of equivariant Eilenberg-MacLane spectra over the cyclic group of order 2 (denoted here by $Q$), so I would like to know all of the examples appearing already in the literature. The ones I am aware of are (I'll list the underlying Mackey functors):

  • the constant Mackey functor $\underline{\mathbb{F}}_2$ - these computations are built on the unpublished work of Stong and appear (among others) in Lewis's article on $RO(Q)$-graded cohomology of complex projective spaces ;
  • the constant Mackey functor $\underline{\mathbb{Z}}$ - its coefficients are computed in Dugger's paper on Atiyah-Hirzebruch Spectral Sequence for $KR$-theory;
  • the Burnside Mackey functor $\underline{A}$ - the description of $RO(Q)$-graded abelian structure is also built on the work of Stong and may also be found in Lewis's paper.

Are there any other examples of such calculations?

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It's done for an arbitrary Mackey functor, for the group $C_p$ for any prime, as Theorem 8.1 here:

https://hopf.math.purdue.edu/Ferland-Lewis/FerlandLewis.pdf

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It's not published yet, but I have a paper on the arXiv that has, in its last section, several other calculations related to $\underline{\mathbb Z}$. There are three other Mackey functors for which the cohomology of a point is a shifted copy of that for $\underline{\mathbb Z}$.

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