This is a somewhat naive question to which I don't know the answer. There is a map of spectra $S \to S$, defined up to homotopy, given by multiplication by $-1$, and it satisfies the relation $(-1)^2 \simeq 1$. Can this be refined to a strict $\mathbb{Z}/2$-action on the sphere $S$ (and thus on all spectra)?

My feeling is that the answer is probably not, but I haven't been able to prove it. I do know that when one inverts $2$, there is a $\mathbb{Z}/2$-action: if I understand correctly, the obstructions to rigidifying a homotopy $\mathbb{Z}/2$-action to an actual $\mathbb{Z}/2$-action on a spectrum $X$ live in the $\mathbb{Z}/2$-cohomology groups of $B Aut(X)$ (where $Aut(X)$ is the monoid of homotopy self-equivalences of $X$), which should be zero when $2$ acts invertibly on $\pi_*X $.