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