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This question is related to the one here, but has a slightly different angle.

Let $G$ be a topological group and let $X$ be a $G$-space. Taking the suspension spectrum $\Sigma^{\infty}_+ X$ (in my setup I am using symmetric spectra, but this is probably not important), I end up with a $\Sigma^{\infty}_+ G$-module spectrum. In case $G$ is the trivial group the suspension spectrum functor is a left adjoint in a Quillen adjunction, where we take the stable model structure on the right hand side (on symmetric spectra in my setup).

Is there a natural model structure on $G$-spaces, such that $\Sigma^{\infty}_+$ becomes a left Quillen functor from $G$-spaces to $\Sigma^{\infty}_+G$-modules? Can I choose this in such a way that the cofibrant objects on the left hand side contain $G$-CW-complexes?

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    $\begingroup$ As a category, $\Sigma^\infty_G$-modules are the same as spectra with a $G$-action. If you want $G$-CW-complexes to be cofibrant, you should take as weak equivalences every map that induces weak equivalences on fixed points for every $H\subset G$. math.rochester.edu/people/faculty/doug/otherpapers/… should give you what you want $\endgroup$
    – Lennart Meier
    Jan 16, 2015 at 13:15

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