  In this article, we prove that if $H$ is a skew field of center $k$ and
$\sigma$ an automorphism of finite order of $H$ such that the fixed subfield
$k^{\langle \sigma \rangle}$ of $k$ under the action of $\sigma$ contains an
ample field, then the inverse Galois problem has a positive answer over the
skew field $H(t,\sigma)$ of twisted rational fractions. Moreover, if
$k^{\langle \sigma \rangle}$ contains either a real closed field, or an
Henselian field of residue characteristic $0$ and containing all roots of
unity, then the profree group of countable rank $\widehat{F}_{\omega}$ is a
Galois group over $H(t,\sigma)$.
