  Let $\Omega$ be a set equipped with an equivalence relation $\sim$; we refer
to the equivalence classes as blocks of $\Omega$. A permutation group $G \le
\mathrm{Sym}(\Omega)$ 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 $k \ge 3$, there are no finite block-faithful
$k$-by-block-transitive actions with nontrivial blocks.
