  In this paper we investigate the cohomological obstruction for the field of
moduli of a $G$-cover to be a field of definition, in the case of local fields
and covers with tame admissible reduction. This applies in particular to
$p$-adic fields where $p$ does not divide the order of the group $G$. We give
examples of $G$-covers with field of moduli $\QQ_p$ that cannot be defined over
$\QQ_p$, for all primes $p>5$.
