Multiple transitivity except for a system of imprimitivity

Let ω be a set equipped with an equivalence relation ∼ \sim; we refer to the equivalence classes as blocks of ω. A permutation group G Sym (ω) G\leq\mathrm{Sym}(\Omega) is -by-block-transitive if ∼ \sim is invariant, with at least blocks, and is transitive on the set of -tuples of points such that no two entries lie in the same block. The action is block-faithful if the action on the set of blocks is faithful. In this article, we classify the finite block-faithful 2-by-block-transitive actions. We also show that, for k ≥ 3 k\geq 3, there are no finite block-faithful -by-block-transitive actions with nontrivial blocks.