Given a field
k
k
and a finite group
G
G
, the Beckmann–Black problem asks whether every Galois field extension
F
/
k
F/k
with group
G
G
is the specialization at some
t
0
∈
k
t_0 \in k
of some Galois field extension
E
/
k
(
T
)
E/k(T)
with group
G
G
and
E
∩
k
¯
=
k
E \cap \overline {k} = k
. We show that the answer is positive for arbitrary
k
k
and
G
G
, if one waives the requirement that
E
/
k
(
T
)
E/k(T)
is normal. In fact, our result holds if
Gal
(
F
/
k
)
\operatorname {Gal}(F/k)
is any given subgroup
H
H
of
G
G
and, in the special case
H
=
G
H=G
, we provide a similar conclusion even if
F
/
k
F/k
is not normal. We next derive that, given a division ring
H
H
and an automorphism
σ
\sigma
of
H
H
of finite order, all finite groups occur as automorphism groups over the skew field of fractions
H
(
T
,
σ
)
H(T, \sigma )
of the twisted polynomial ring
H
[
T
,
σ
]
H[T, \sigma ]
.