On embedding problems with restricted ramifications

Let L/k be a Galois extension with Galois group G, and (epsilon) : 1 --> A --> E --> G --> 1 a central extension. We study the existence of the Galois extension M/L/k such that the Galois group Gal(M/k) is isomorphic to E and that MIL is unramified outside S, where S is a finite set of primes of L. As an application, we also study the class number of the Hilbert p-class fi eld.