  Let $G$ be a group acting on a small category $\mathcal C$ over a field $k$,
that is $\mathcal C$ is a $G$-$k$-category. We first obtain that $\mathcal C$
is resolvable by a category which is $G$-$k$-equivalent to it, on which $G$
acts freely on objects.
  This resolvent category enables to show that if the coinvariants and the
invariants functors are exact, then the coinvariants and invariants of the
Hochschild-Mitchell (co)homology of $\mathcal C$ are isomorphic to the trivial
component of the Hochschild-Mitchell (co)ho\-mo\-logy of the skew category
$\mathcal C[G]$. Otherwise the corresponding spectral sequence can be settled.
  If the action of $G$ is free on objects, there is a canonical decomposition
of the Hochschild-Mitchell (co)homology of the quotient category $\mathcal C/G$
along the conjugacy classes of $G$. This way we provide a general frame for
monomorphisms which have been described previously in low degrees.
