Let $E$ be a spectrum.  Then $E \wedge E$ is a $\mathbb{Z}/2$-spectrum in the naivest possible sense, i.e., an object with $\mathbb{Z}/2$-action in the (∞,1)-category of spectra.  Can I make it a $\mathbb{Z}/2$-spectrum in the less naive, but still not genuine, sense?  (That is, a $\mathbb{Z}/2$-spectrum indexed on the trivial universe.)

I'm thinking of something like the following.  I may represent $E$ as an (reduced & continuous) excisive functor from pointed spaces to pointed spaces.  Then define

$$G(X) = \mathrm{colim}_{I \times I} \mathrm{Map}(S^{x_1} \wedge S^{x_2}, E(S^{x_1}) \wedge E(S^{x_2}) \wedge X)$$

where $I$ is the category of finite sets and inclusions.  Hopefully $G$ is a functor from spaces to $\mathbb{Z}/2$-spaces.  If I forget about the $\mathbb{Z}/2$-fixed point set, I can think of it as $E \wedge E$ with its $\mathbb{Z}/2$ action.  What spectrum does $G(X)^{\mathbb{Z}/2}$ correspond to?  Is there a more familiar name for it?  **Edit:** I seem to be getting $E \vee (E \wedge E)^{h\mathbb{Z}/2}$, but without much confidence.

[Leftover part of the question: If so, by my question <a href="https://mathoverflow.net/questions/3154/1-categorical-description-of-equivariant-homotopy-theory">here</a> I can think of the resulting object as a functor from the opposite of the orbit category of $\mathbb{Z}/2$ to spectra.  Unpacking this amounts to giving some spectrum $F$ together with a map $F \to (E \wedge E)^{h\mathbb{Z}/2}$.  What is $F$?]