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 𝑘-by-block-transitive if ∼ \sim is 𝐺-invariant, with at least 𝑘 blocks, and 𝐺 is transitive on the set of 𝑘-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 𝑘-by-block-transitive actions with nontrivial blocks.