 Solution

For $1\le t\le |\Omega |$, a group is sharply t-transitive when any two ordered $t$-tuples of distinct points are related by exactly one group element. Equivalently its action on the set of such tuples is regular. In the finite caseand the stabilizer subgroup of an ordered $t$-tuple is trivial. The condition includes both existence and uniqueness, not just transitivity.