  For a finite Galois extension of fields L/k with Galois group G, we study a
functor from the G-equivariant stable homotopy category to the stable motivic
homotopy category over k induced by the classical Galois correspondence. We
show that after completing at a prime and eta (the motivic Hopf map) this
results in a full and faithful embedding whenever k is real closed and L =
k[i]. It is a full and faithful embedding after eta-completion if a motivic
version of Serre's finiteness theorem is valid. We produce strong necessary
conditions on the field extension L/k for this functor to be full and faithful.
Along the way, we produce several results on the stable C_2-equivariant Betti
realization functor and prove convergence theorems for the p-primary
C_2-equivariant Adams spectral sequence.
