  Given a Hilbertian field $k$ and a finite set $\mathcal{S}$ of Krull
valuations of $k$, we show that every finite split embedding problem $G
\rightarrow {\rm{Gal}}(L/k)$ over $k$ with abelian kernel has a solu\-tion
${\rm{Gal}}(F/k) \rightarrow G$ such that every $v \in \mathcal{S}$ is totally
split in $F/L$. Two applications are then given. Firstly, we solve a
non-constant variant of the Beckmann--Black problem for solvable groups: given
a field $k$ and a non-trivial finite solvable group $G$, every Galois field
extension $F/k$ of group $G$ is shown to occur as the specialization at some
$t_0 \in k$ of some Galois field extension $E/k(T)$ of group $G$ with $E \not
\subseteq \overline{k}(T)$. Secondly, we contribute to inverse Galois theory
over division rings, by showing that, for every division ring $H$ and every
automorphism $\sigma$ of $H$ of finite order, all finite semiabelian groups
occur as Galois groups over the skew field of fractions $H(T, \sigma)$ of the
twisted polynomial ring $H[T, \sigma]$.
