Embedding problem

Given a field K and a finite group H, one may pose the following question (the so called inverse Galois problem).

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.

Finite embedding problems characterize profinite groups.

Let F be a countably (topologically) generated profinite group.