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The "group ring spectrum" $SG$ you ask for does indeed exist, but modules over it are not the same as $G$-spectra. Indeed, $SG$ is just the suspension spectrum $\Sigma^\infty_+ G$ of $G$ as a discrete space. The suspension spectrum of any $A_\infty$ space is naturally an $A_\infty$ ring spectrum, so the group structure on $G$ gives an $A_\infty$ structure on $SG$. An $SG$-module can be shown to be the same thing as a spectrum with a (coherent) action of $G$.

Any $G$-spectrum has an "underlying" non-equivariant spectrum which has the natural structure of an $SG$-module. However, this is not an equivalence of categories, and the basic reason is that they correspond to different notions of "weak equivalence" of equivariant objects. For simplicity, I'll describe this in the unstable setting of $G$-spaces. If $X$ and $Y$ are two spaces with an action of a group $G$ and $f:X\to Y$ is an equivariant map, there are two things we might mean when we say $f$ is an "equivariant homotopy equivalence". The first is that $f$ is an ordinary homotopy equivalence of the spaces $X$ and $Y$ which happens to also be an equivariant map. The second is that $f$ has a homotopy inverse internal to the category of $G$-spaces: there exists another equivariant map $g:Y\to X$ such that the compositions $fg$ and $gf$ are homotopy to the identity through equivariant maps. This notion is much stronger. For example, the map $EG\to *$ is a homotopy equivalence in the weaker sense, but not in this stronger sense, because $EG$ has no fixed points so there are no equivariant maps $*\to EG$. If we restrict to $G$-CW complexes (a natural equivariant generalization of CW-complexes), it turns out that a map $X\to Y$ is an equivariant equivalence in this stronger sense iff for every subgroup $H\subseteq G$, the induced map $X^H\to Y^H$ on fixed points is a homotopy equivalence.

The category of $SG$-modules is a stable version of $G$-spaces under the first, weaker notion of equivalence. Indeed, as with any ring spectrum, a weak equivalence of $SG$-modules is just a map of $SG$-modules which happens to be a weak equivalence of underlying spectra (though an inverse which is actually an $SG$-module map can be found if we're willing to take cofibrant and fibrant replacements of our modules, which corresponds to replacing a $G$-space $X$ with the Borel construction $EG \times_G times X$). On the other hand, the category of $G$-spectra is a stable version of $G$-spaces under the second, stronger notion of equivalence.
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The "group ring spectrum" $SG$ you ask for does indeed exist, but modules over it are not the same as $G$-spectra. Indeed, $SG$ is just the suspension spectrum $\Sigma^\infty_+ G$ of $G$ as a discrete space. The suspension spectrum of any $A_\infty$ space is naturally an $A_\infty$ ring spectrum, so the group structure on $G$ gives an $A_\infty$ structure on $SG$. An $SG$-module can be shown to be the same thing as a spectrum with a (coherent) action of $G$.

Any $G$-spectrum has an "underlying" non-equivariant spectrum which has the natural structure of an $SG$-module. However, this is not an equivalence of categories, and the basic reason is that they correspond to different notions of "weak equivalence" of equivariant objects. For simplicity, I'll describe this in the unstable setting of $G$-spaces. If $X$ and $Y$ are two spaces with an action of a group $G$ and $f:X\to Y$ is an equivariant map, there are two things we might mean when we say $f$ is an "equivariant homotopy equivalence". The first is that $f$ is an ordinary homotopy equivalence of the spaces $X$ and $Y$ which happens to also be an equivariant map. The second is that $f$ has a homotopy inverse internal to the category of $G$-spaces: there exists another equivariant map $g:Y\to X$ such that the compositions $fg$ and $gf$ are homotopy to the identity through equivariant maps. This notion is much stronger. For example, the map $EG\to *$ is a homotopy equivalence in the weaker sense, but not in this stronger sense, because $EG$ has no fixed points so there are no equivariant maps $*\to EG$. If we restrict to $G$-CW complexes (a natural equivariant generalization of CW-complexes), it turns out that a map $X\to Y$ is an equivariant equivalence in this stronger sense iff for every subgroup $H\subseteq G$, the induced map $X^H\to Y^H$ on fixed points is a homotopy equivalence.

The category of $SG$-modules is a stable version of $G$-spaces under the first, weaker notion of equivalence. Indeed, as with any ring spectrum, a weak equivalence of $SG$-modules is just a map of $SG$-modules which happens to be a weak equivalence of underlying spectra (though an inverse which is actually an $SG$-module map can be found if we're willing to take cofibrant and fibrant replacements of our modules, which corresponds to replacing a $G$-space $X$ with the Borel construction $EG \times_G X$). On the other hand, the category of $G$-spectra is a stable version of $G$-spaces under the second, stronger notion of equivalence.