  For a global field, local field, or finite field $k$ with infinite Galois
group, we show that there can not exist a functor from the Morel--Voevodsky
$\mathbb{A}^1$-homotopy category of schemes over $k$ to a genuine Galois
equivariant homotopy category satisfying a list of hypotheses one might expect
from a genuine equivariant category and an \'etale realization functor. For
example, these hypotheses are satisfied by genuine $\mathbb{Z}/2$-spaces and
the $\mathbb{R}$-realization functor constructed by Morel--Voevodsky. This
result does not contradict the existence of \'etale realization functors to
(pro-)spaces, (pro-)spectra or complexes of modules with actions of the
absolute Galois group when the endomorphisms of the unit is not enriched in a
certain sense. It does restrict enrichments to representation rings of Galois
groups.
