For Segal's complex $G$-equivariant $K$-theory, it is well-known that there is an Atiyah-Hirzebruch spectral sequence. If say $G$ a finite group and $X$ a finite CW-complex, the second page of this spectral sequence can be identified with Bredon equivariant cohomology $H^\bullet_G$ with coefficients in the representation ring. More precisely, there is a spectral sequence $E^{p,q}_r$ converging to $K^{p+q}_G(X)$ with $$ E^{p,q}_2= \begin{cases} H^p_G(X,\mathcal{R}_G) & q \text{ even,} \\ 0 & q \text{ odd}. \end{cases} $$ The coefficient functor $\mathcal{R}_G:\mathcal{O}_G→\mathbf{Ab}$ from the orbit category of $G$ to abelian groups is the ring of virtual complex representations on objects and can be defined on morphisms by restricting and conjugating representations. This tool is really nice to compute easy examples.

I am aware that complex $K$-theory is much nicer than $KR$-theory and that $KR$-theory is something different from $\mathbb{Z}_2$-equivariant $K$-theory, since in $KR$-theory the $\mathbb{Z}_2$ acts complex anti-linearly. But I am very much tempted to say that that there is a spectral sequence $E^{p,q}_r$ converging to $KR^{p+q}(X)$ with $$ E^{p,q}_2=H^p_{\mathbb{Z}_2}(X,\mathcal{KR}^q), $$ where $\mathcal{KR}^q:\mathcal{O}_{\mathbb{Z}_2}→\mathbf{Ab}$ is some functor that assigns $KR^q(pt)$ to a point $pt=\mathbb{Z}_2/\mathbb{Z}_2$ and $$ KR^q(\mathbb{Z}_2)=KR^q(S^{1,0})≅K^q(pt) $$ to the homogeneous space $\mathbb{Z}_2$. I would not be surprised if the general arguments of Segal's paper on spectral sequences http://www.maths.ed.ac.uk/~v1ranick/papers/segalclass.pdf would apply here, yielding something like the above. So I think we should be able to compute $KR$-theory by computing $\mathbb{Z}_2$-equivariant Bredon cohomology with certain coefficients.

My questions now are: Does my naive spectral sequence work? If it does, why are the spectral sequences I find when I use google so complicated? If it does not work, why not? Do I need to consider the theory of $RO(G)$-graded cohomology theories or $G$-spectra? I must admit that I am not very familiar with the theory of spectra, let alone $G$-spectra, so there could be something missing there. Is there another spectral sequence that computes $KR$-theory of finite CW-complexes (or even better: equivariant $KR_G$-theory) with second page in a version of Bredon equivariant cohomology?

Thank you very much for your help.

For Segal's complex $G$-equivariant $K$-theory, it is well-known that there is an Atiyah-Hirzebruch spectral sequence. If say $G$ a finite group and $X$ a finite CW-complex, the second page of this spectral sequence can be identified with Bredon equivariant cohomology $H^\bullet_G$ with coefficients in the representation ring. More precisely, there is a spectral sequence $E^{p,q}_r$ converging to $K^{p+q}_G(X)$ with $$ E^{p,q}_2= \begin{cases} H^p_G(X,\mathcal{R}_G) & q \text{ even,} \\ 0 & q \text{ odd}. \end{cases} $$ The coefficient functor $\mathcal{R}_G:\mathcal{O}_G→\mathbf{Ab}$ from the orbit category of $G$ to abelian groups is the ring of virtual complex representations on objects and can be defined on morphisms by restricting and conjugating representations. This tool is really nice to compute easy examples.

I am aware that complex $K$-theory is much nicer than $KR$-theory and that $KR$-theory is something different from $\mathbb{Z}_2$-equivariant $K$-theory, since in $KR$-theory the $\mathbb{Z}_2$ acts complex anti-linearly. But I am very much tempted to say that that there is a spectral sequence $E^{p,q}_r$ converging to $KR^{p+q}(X)$ with $$ E^{p,q}_2=H^p_{\mathbb{Z}_2}(X,\mathcal{KR}^q), $$ where $\mathcal{KR}^q:\mathcal{O}_{\mathbb{Z}_2}→\mathbf{Ab}$ is some functor that assigns $KR^q(pt)$ to a point $pt=\mathbb{Z}_2/\mathbb{Z}_2$ and $$ KR^q(\mathbb{Z}_2)=KR^q(S^{1,0})≅K^q(pt) $$ to the homogeneous space $\mathbb{Z}_2$. I would not be surprised if the general arguments of Segal's paper on spectral sequences http://www.maths.ed.ac.uk/~v1ranick/papers/segalclass.pdf would apply here, yielding something like the above. So I think we should be able to compute $KR$-theory by computing $\mathbb{Z}_2$-equivariant Bredon cohomology with certain coefficients.

Asking google for a spectral sequence to compute $KR$-theory results in this paper by Dugger: https://arxiv.org/pdf/math/0304099.pdf Moreover, the following master thesis seems to be based on Dugger's work: http://www2.math.uni-wuppertal.de/~hornbost/markett.pdf

My questions now are: Does my naive spectral sequence work? If it does, why are the spectral sequences I find when I use google so complicated? Notably, If it does not work, why not? Do I need to consider the theory of $RO(G)$-graded cohomology theories or $G$-spectra? I must admit that I am not very familiar with the theory of spectra, let alone $G$-spectra, so there could be something missing there. Is there another spectral sequence that computes $KR$-theory of finite CW-complexes (or even better: equivariant $KR_G$-theory) with second page in a version of Bredon equivariant cohomology?

Thank you very much for your help.