  Let $G$ be a group acting on a finite set $\Omega$. Then $G$ acts on
$\Omega\times \Omega$ by its entry-wise action and its orbits form the basis
relations of a coherent configuration (or shortly scheme). Our concern is to
consider what follows from the assumption that the number of orbits of $G$ on
$\Omega_i\times \Omega_j$ is constant whenever $\Omega_i$ and $\Omega_j$ are
orbits of $G$ on $\Omega$. One can conclude from the assumption that the
actions of $G$ on ${\Omega_i}$'s have the same permutation character and are
not necessarily equivalent. From this viewpoint one may ask how many
inequivalent actions of a given group with the same permutation character there
exist. In this article we will approach to this question by a purely
combinatorial method in terms of schemes and investigate the following topics:
  (i) balanced schemes and their central primitive idempotents, (ii)
characterization of reduced balanced schemes.
