This question is related to the one <a href="http://mathoverflow.net/questions/80163/are-g-spectra-the-same-as-modules-over-a-group-ring-spectrum">here</a>, 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?