The universally solvable embedding problem with cyclic kernel (Q1866700)

Let \(1\to A\to G@>\varphi>> F\to 1\) be an exact sequence of finite groups. One says that it determines a universally solvable embedding problem if the embedding problems associated with this sequence are solvable for any pair of fields \(K| k\), where \(K\) is a Galois extension of \(k\) with Galois group \(F\). The authors prove that the universally solvable embedding problems connected with the sequence above is semidirect if \(A\) is cyclic.