After an introduction concerned with equivalence relations and the tower associated to them, the authors study some properties of \(G\)-relations (where \(G\) is a group) and connect them with the algebras \(A_n^G(d)\) (algebras that have a basis indexed by ``\(G\)-stable equivalence relations''). Then it is studied the tower \(\{A_n^G(d):n\geq 1\}\) where \(G\) is a finite group acting free on a set and has \(2n\) orbits, \(d\) is a complex parameter. It is proved that for almost all values of \(d\), \(A_n^G(d)\) is semi-simple and, for the `generic case', it is determined the representation theory and the Bratteli diagram of the associated tower.