  The De Concini-Procesi wonderful models of the braid arrangement of type
$A_{n-1}$ are equipped with a natural $S_n$ action, but only the minimal model
admits an `hidden' symmetry, i.e. an action of $S_{n+1}$ that comes from its
moduli space interpretation. In this paper we explain why the non minimal
models don't admit this extended action: they are `too small'. In particular we
construct a {\em supermaximal} model which is the smallest model that can be
projected onto the maximal model and again admits an extended $S_{n+1}$ action.
We give an explicit description of a basis for the integer cohomology of this
supermaximal model.
  Furthermore, we deal with another hidden extended action of the symmetric
group: we observe that the symmetric group $S_{n+k}$ acts by permutation on the
set of $k$-codimensionl strata of the minimal model. Even if this happens at a
purely combinatorial level, it gives rise to an interesting permutation action
on the elements of a basis of the integer cohomology.
