Let R be the 2-periodic complex K-theory spectrum, or any other naturally occuring 2-periodic E-infty ring spectrum. The suspend-once functor gives an autoequivalence of the category of R-module spectra, and since R is 2-periodic applying it twice is isomorphic to the identity functor. In some coarse sense this defines an action of Z/2 on the category of R-modules. 


There is a better, more complicated and more correct notion of Z/2-action on a collection of module spectra. Module spectra form an infty-category, and we can ask if Z/2 acts on this infty-category. One way to put it: the suspend-once functor and the homotopy between suspend-twice and the identity give us a map from RP^2 into the classifying space of automorphisms of this infty-category. Does this extend to a map from BZ/2 = RP^infty?



How often does suspension define an action of Z/2 on a category of module spectra?
At least for complex K-theory, I believe the answer is yes. If you want an explicit
construction, you can use the fact that there's a symmetric monoidal functor from 
the 2-groupoid of Clifford algebras and Morita equivalences to the groupoid of invertible K module spectra. So it suffices to construct a monoidal functor from Z/2 into this 2-groupoid, which is a much more concrete problem. 


