  Let $p$ be a prime number and let $X$ be a complex algebraic variety with an
action of $\mathbb{Z}/p\mathbb{Z}$. We develop the theory of parity complexes
in a certain $2$-periodic localization of the equivariant constructible derived
category $D^b_{\mathbb{Z}/p\mathbb{Z}}(X,\mathbb{Z}_p)$. Under certain
assumptions, we use this to define a functor from the category of parity
sheaves on $X$ to the category of parity sheaves on the fixed-point locus
$X^{\mathbb{Z}/p\mathbb{Z}}$. This may be thought of as a categorification of
Smith theory. When $X$ is the affine Grassmannian associated to some complex
reductive group, our functor gives a geometric construction of the
Frobenius-contraction functor recently defined by M. Gros and M. Kaneda via the
geometric Satake equivalence.
