Rigidity and exotic models for _v1-local G-equivariant stable homotopy

We prove that the v1-local G-equivariant stable homotopy category for G
a finite group has a unique G-equivariant model at p = 2. This means that at the
prime 2 the homotopy theory of G-spectra up to fixed point equivalences on K-theory is uniquely determined by its triangulated homotopy category and basic Mackey structure. The result combines the rigidity result for K-local spectra of the second author with the equivariant rigidity result for G-spectra of the first author. Further, when the prime p is at least 5 and does not divide the order of G, we provide an algebraic exotic model as well as a G-equivariant exotic model for the v1-local G-equivariant stable homotopy category, showing that for primes p ≥ 5 equivariant rigidity fails in general.