Embedding problem

In Galois theory, a branch of mathematics, the embedding problem is a generalization of the inverse Galois problem. Roughly speaking, it asks whether a given Galois extension can be embedded into a Galois extension in such a way that the restriction map between the corresponding Galois groups is given.
Definition.
Given a field "K" and a finite group "H", one may pose the following question (the so called inverse Galois problem). Is there a Galois extension "F/K" with Galois group isomorphic to "H". The embedding problem is a generalization of this problem:
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"?
Analogously, an embedding problem for a profinite group "F" consists of the following data: Two profinite groups "H" and "G" and two continuous epimorphisms "φ" : "F" → "G" and
"f" : "H" → "G". The embedding problem is said to be finite if the group "H" is.
A solution (sometimes also called weak solution) of such an embedding problem is a continuous homomorphism "γ" : "F" → "H" such that "φ" = "f" "γ". If the solution is surjective, it is called a proper solution.
Properties.
Finite embedding problems characterize profinite groups. The following theorem gives an illustration for this principle.
Theorem. Let "F" be a countably (topologically) generated profinite group. Then 