at.algebraic topology - Naive Z/2-spectrum structure on E smash E?

Given a spectrum $E$ there is a "standard" lift of $E wedge E$ to a $mathbb{Z}/2$-spectrum using the basic technique you describe. One way to describe it as follows.

You can construct the category of genuine $mathbb{Z}/2$-spectra (indexed on the full universe) via collections of $mathbb{Z}/2$-spaces $X_n$ with equivariant structure maps $sigma: S^V wedge X_n to X_{n+1}$, where $S^V = S^1 wedge S^1$ is the 1-point compactification of the regular representation $mathbb{R} times mathbb{R}$ with the "flip" action. Under this description, if $E$ is a spectrum made up of spaces $E_n$ and structure maps $S^1 wedge E_n to E_{n+1}$, then you can construct $E wedge E$ as a genuine $mathbb{Z}/2$-spectrum with spaces $E_n wedge E_n$ and structure maps $S^1 wedge S^1 wedge E_n wedge E_n$ that simply twist and apply the structure map on each factor. (This, e.g., is one way to pass forward the equivariant structure on $TC$).

This fully genuine spectrum has an underlying spectrum indexed on the trivial subgroup. The $mathbb{Z}/2$-fixed point object is the homotopy pullback of a diagram

$$
(E wedge E)^{hmathbb{Z}/2} to (E wedge E)^{tmathbb{Z}/2} leftarrow E
$$

where $Z^{tmathbb{Z}/2}$ is the so-called "Tate spectrum" of $Z$, which is to Tate cohomology as the homotopy fixed point spectrum is to group cohomology.

If $E = Sigma^infty W$ for a space $W$ then the map from $E$ to the Tate spectrum lifts (via the diagonal) to a map to the homotopy fixed point spectrum, and so the homotopy pullback will actually be homotopy equivalent to $E vee (E wedge E)_{hmathbb{Z}/2}$. (This homotopy orbit is the fiber of the map from homotopy fixed points to Tate fixed points.)

For finite spectra the map from $E$ to its Tate spectrum is 2-adic completion; this is one way to state the content of the Segal conjecture that Carlsson proved (at least at the prime 2). Sverre Lunoe-Nielsen extended this result to a number of other spectra like the Brown-Peterson spectra. In these cases the fixed-point object is equivalent after 2-adic completion to the homotopy fixed point object.

All the above plays out the same way for a cyclic group of prime order.