In both the orthogonal world and the EKMM world one can set things up so that $G$-spectra are just spectra with an action of $G$, and the naive and genuine equivariant stable categories are the homotopy categories for two different model structures on the same underlying geometric category. I will take that point of view.
I will write $\kappa^G(X)$ for the categorical fixed points of a spectrum $X$ with $G$-action, i.e. the largest subspectrum on which the action is trivial. We also have the Lewis-May fixed point functor which I will denote by $\lambda^G(\cdot)$, and the geometric fixed point functor $\phi^G(\cdot)$.
Here are some facts about the EKMM category:
- The functor $\Sigma^\infty$ preserves smash products on the nose, but $\Sigma^\infty X$ is not cofibrant unless $X$ is a point.
- The functor $\Sigma^\infty$ also commutes with $\kappa^G$, so $\kappa^{C_2}(X\wedge X)=X$ whenever $X$ is a suspension spectrum.
- Now let $T$ be the cofibrant replacement of $S^0$. This is an $\mathbb{L}$-spectrum in the sense of EKMM, so it has a kind of action of the monoid $\mathcal{L}(1)=\mathcal{L}(\mathcal{U},\mathcal{U})$ of linear isometric embeddings of the universe $\mathcal{U}=\mathbb{R}^\infty$ in itself. If we just consider the underlying Lewis-May spectrum, it can be identified with $\Sigma^\infty\mathcal{L}(\mathcal{U},\mathcal{U})_+$. Similarly, $T\wedge T$ is just $\Sigma^\infty\mathcal{L}(\mathcal{U}^2,\mathcal{U})_+$. From this it is easy to see that $\kappa^{C_2}(T\wedge T)=0$.
- Similarly, I am pretty sure that $\kappa^{C_2}(X\wedge X)=0$ whenever $X$ is cofibrant.
I think that the story in the orthogonal world is similar, but the analysis is a bit more complicated and I do not have the details to hand.