We study the inverse Galois problem with restricted
ramification. Let $K$ be an algebraic number field and $G$ be
a $2$-group. We consider the question whether there exists an
unramified Galois extension $M/K$ with Galois group isomorphic
to $G$. We study this question using the theory of embedding
problems. Let $L/k$ be a Galois extension and $(\varepsilon):
1\to \mathbf{Z}/2\mathbf{Z}\to E\to \operatorname{Gal}
(L/k)\to 1$ a central extension. We first investigate the
existence of a Galois extension $M/L/k$ such that the Galois
group $\operatorname{Gal} (M/k)$ is isomorphic to $E$ and any
finite prime is unramified in $M/L$. As an application, we
prove the existence of an unramified extension over cyclic
quintic fields with Galois group isomorphic to
$32{\Gamma}_5a_2$ under the condition that the class number is
even. We also consider the Fontaine-Mazur-Boston Conjecture in
the case of abelian $l$-extensions over $\mathbf{Q}$.