Galois group
Let "L/K" be a Galois extension with Galois group "G" and let "f" : "H" → "G" be an epimorphism. Is there a Galois extension "F/K" with Galois group "H" and an embedding "α" : "L" → "F" fixing "K" under which the restriction map from the Galois group of "F/K" to the Galois group of "L/K" coincides with "f"?