Symmetric graphs with complete quotients

Let X be a G-symmetric graph of valency s, with vertex set V. We suppose that V admits a G-partition P, with blocks of size v, and that the quotient graph of X modulo P induced on the vertex set P is a complete graph of order b+1. Then, for each pair of distinct blocks B, C from P, the graph [B,C] induced on the union B\cup C is bipartite with each vertex of valency 0 or t. When t=1, we showed in earlier papers how the 1-design D(B) induced on the block B can sometimes be used to classify possible triples (X, G, P). Here we extend these ideas to t > 1 and prove that, if the group induced by G on a block B is 2-transitive and the "blocks" of D(B) have size k < v, then either (i) v < b, or (ii) v is at least b and the triple (X, G, P) occurs in a given list. This extends an earlier result which classified triples (X, G, P) with induced 2-transitive action on a block and v > b+1, when t = 1 and k = v.