For a finite group G we study certain rings called k-S-rings, one for each non-negative integer k, where the 1-S-ring is the centralizer ring of G. These rings have the property that the (k+1)-S-ring determines the k-S-ring. We show that the 4-S-ring determines G when G is any group with finite classes. We show that the 3-S-ring determines G for any finite group G, thus giving an answer to a question of Brauer. We show the 2-characters defined by Frobenius and the extended 2-characters of Ken Johnson are characters of representations of the 2-S-ring of G. We find the character table for the 2-S-ring of the dihedral groups of order 2n, n odd, and classify groups with commutative 3-S-ring.