  In this paper, we develop the theory of equivariant motivic homotopy theory,
both unstable and stable. While our original interest was in the case of
profinite group actions on smooth schemes, we discuss our results in as broad a
setting as possible so as to be applicable in a variety of contexts, for
example to the case of smooth group scheme actions on schemes that are not
necessarily smooth. We also discuss how ${\mathbb A}^1$-localization behaves
with respect to mod-$\ell$-completion, where $\ell$ is a fixed prime.
