Structure of the Galois group of the maximal unramified pro-2-extension of some Z(2)-extensions

For a number field k, we consider the Galois group G = Gal(L(k(infinity))/k(infinity)) of the maximal unramified pro-2-extension of the cyclotomic Z(2)-extension k(infinity) of k. In terms of transfer, we establish a necessary and sufficient condition for a 2-group to be abelian or metacyclic non-abelian whenever its abelianization is of type (2(n), 2(m)), with n >= 2 and m >= 2. Then we apply this result to construct an infinite family of real quadratic fields for which G is an abelian pro-2-group of rank 2.