 Free actions on even-dimensional spheres have order at most two

For every nonidentity element of a free group action on $S^{2r}$, its action map has no fixed point and hence degree minus one by fixed-point-free sphere maps are homotopic to the antipodal map. Degree of homeomorphisms defines a homomorphism into $\{1,-1\}$, whose kernel is trivial, since no nonidentity action map can have degree one. Thus the acting group has order at most two. For $S^0$, freeness on its two points gives the same conclusion directly.