Explicit forms of local lifting for GL₂

Let F be a local non-Archimedean field and let S(GL₂(F)) be the set of equivalence classes of irreducible admissible representations of GL₂(F). When K/F be a Galois field extension, there is a map, called lifting, from S(GL₂(F)) to S(GL₂(K)). We give an explicit form of lifting when K/F is a quadratic wildly ramified extension and the given representations are Weil supercuspidal. We also provide a comparison between Weil representations and induced representations of GL₂(F).