Let $X$ be an object in one of the well-known symmetric monoidal model categories of spectra. E.g., an $\mathtt S$-module in the sense of EKMM, or an orthogonal spectrum, or a $\Gamma$-space, etc.

One can form the spectrum $X\wedge X$, which is a "naive" spectrum with an action of $\Sigma_2$, and one can take the categorical fixed points to obtain the spectrum $(X\wedge X)^{\Sigma_2}$. What can be said about the homotopy type of this spectrum as a functor of $X$?

More specifically I would like to know the following:

• Does this construction have good homotopical properties. For example, does it preserve weak equivalences between cofibrant spectra?
• Does the homotopy type of $(X\wedge X)^{\Sigma_2}$ (say for cofibrant $X$) depend on the specific model of smash product?
• Is it true that if $X$ is a suspension spectrum then $(X\wedge X)^{\Sigma_2}\simeq X$? If it is not always true, is it true in some cases, i.e., for some models of smash product?
• Can the homotopy type of $(X\wedge X)^{\Sigma_2}$ be described in other cases?