A reciprocity law for certain Frobenius extensions

Let E/F be a finite Galois extension of algebraic number fields with Galois group G. Assume that G is a Frobenius group and H is a Frobenius complement of G. Let F(H) be the maximal normal nilpotent subgroup of H. If H/F(H) is nilpotent, then every Artin L-function attached to an irreducible representation of G arises from an automorphic representation over F, i.e., the Langlands' reciprocity conjecture is true for such Galois extensions.