Let
G
G
be a finite group and
K
K
a number field. We construct a
G
G
-extension
E
/
F
E/F
, with
F
F
of transcendence degree
2
2
over
K
K
, that specializes to all
G
G
-extensions of
K
p
K_\mathfrak {p}
, where
p
\mathfrak {p}
runs over all but finitely many primes of
K
K
. If furthermore
G
G
has a generic extension over
K
K
, we show that the extension
E
/
F
E/F
has the so-called Hilbert–Grunwald property. These results are compared to the notion of essential dimension of
G
G
over
K
K
, and its arithmetic analogue.