I came across the following claim in "Adventures in Group Theory" by David Joyner that I couldn't find a proof for.
If $k>5$ and $G$ is a group acting $k$-transitively on a finite set $X$ then $G$ is isomorphic to $S_m$ or to $A_n$, for some $m \ge k$ or some $n \ge k+2$.
If $k \le 5$ what are the counter examples?