  We study solutions to the Brauer embedding problem with restricted
ramification. Suppose $G$ and $A$ are a abelian groups, $E$ is a central
extension of $G$ by $A$, and
$f:\text{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})\rightarrow G$ a continuous
homomorphism. We determine conditions on the discriminant of $f$ that are
equivalent to the existence of an unramified lift
$\widetilde{f}:\text{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})\rightarrow E$ of
$f$.
  As a consequence of this result, we use conditions on the discriminant of $K$
for $K/\mathbf{Q}$ abelian to classify and count unramified nonabelian
extensions $L/K$ normal over $\mathbf{Q}$ where the (nontrivial) commutator
subgroup of $\text{Gal}(L/\mathbf{Q})$ is contained in its center. This
generalizes a result due to Lemmermeyer, which states that a quadratic field
$\mathbf{Q}(\sqrt{d})$ has an unramified extension normal over $\mathbf{Q}$
with Galois group $H_8$ the quaternion group if and only if the discriminant
factors $d=d_1 d_2 d_3$ as a product of three coprime discriminants, at most
one of which is negative, satisfying the following condition on Legendre
symbols: \[ \left(\frac{d_i d_j}{p_k}\right)=1 \] for $\{i,j,k\}=\{1,2,3\}$ and
$p_i$ any prime dividing $d_i$.
