Certain Unramified Metabelian Extensions Using Lemmermeyer Factorizations

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 $f:\text{Gal}(\bar{\mathbf{Q}}\mathrm{/}\mathbf{Q})\to G$ a continuous homomorphism. We determine conditions on the discriminant of $f$ that are equivalent to the existence of an unramified lift $\tilde{f}:\text{Gal}(\bar{\mathbf{Q}}\mathrm{/}\mathbf{Q})\to E$ of $f$. As a consequence of this result, we use conditions on the discriminant of $K$ for $K\mathrm{/}\mathbf{Q}$ abelian to classify and count unramified nonabelian extensions $L\mathrm{/}K$ normal over $\mathbf{Q}$ where the (nontrivial) commutator subgroup of $\text{Gal}(L\mathrm{/}\mathbf{Q})$ 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 $d_i$.