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.)

If so, by my question <a href="http://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 $F$?