  A theorem of Grothendieck tells us that if the Galois action on the Tate
module of an abelian variety factors through a smaller field, then the abelian
variety, up to isogeny and finite extension of the base, is itself defined over
the smaller field. Inspired by this, we give a Galois descent datum for a
motive $H$ over a field by asking that the Galois action on an $\ell$-adic
realisation factor through a smaller field. We conjecture that this descent
datum is effective, that is if a motive $H$ satisfies the above criterion, then
it must itself descend to the smaller field.
  We prove this conjecture for K3 surfaces, under some hypotheses. The proof is
based on Madapusi-Pera's extension of the Kuga-Satake construction to arbitrary
characteristic.
