= Solution
If $G$ is $n$-transitive, it is transitive by extending one-element tuples to $n$-tuples. Apply $n$-transitivity to tuples beginning with $\omega$ to see that the <stabilizer subgroup> $G_\omega$ is $(n-1)$-transitive on the remaining elements.
Conversely, given two ordered $n$-tuples of distinct elements, transitivity first supplies $g$ carrying their first entries to each other. Conjugating the given stabilizer hypothesis shows that the stabilizer of the target first entry is $(n-1)$-transitive. An element of that stabilizer then maps the remaining entries of the image tuple to their targets. Composing the two elements proves \b[$n$-transitivity]. For $n=1$, the condition on zero-tuples is vacuous.
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