Serre's conjecture and base change for GL(2)

Taking a totally real finite Galois extension F/Q with Galois group G, we study a zero-dimensional automorphic variety S coming from a definite quaternion algebra B over F which descends to a quaternion algebra B₀ over Q. We state an absolutely elementary conjecture on the (permutation) action of G on S. This conjecture is `basically' equivalent to the existence of base change lift of holomorphic elliptic modular forms (of weight 2) relative to F/Q, assuming Serre's conjecture on modularity of two-dimensional mod p Galois representations of the absolute Galois group.