If $G$ is $k$-transitive but not $(k+1)$-transitive, is it true that $G$ is sharply $k$-transitive?

Question

There is an exercise:

If $G$ is $k$-transitive but not $(k+1)$-transitive, is it true that $G$ is sharply $k$-transitive?

I solved this exercise:

If $G$ is sharply $k$-transitive then $G$ is not $(k+1)$-transitive.

I try to prove the first exercise. But I don't have any idea how to solve it. I don't know if it's true or false. I think it's false but which counterexample?

Any kind of suggestion is appreciated. Thanks to everyone for the help.

Answer 1

In fact, there is a counterexample among Mathieu groups (for instance $M_{24}$ is a $5$-transitive group of degree $24$ which is easily seen to be neither $6$-transitive nor sharply $5$-transitive), but there is a way simpler example.

Consider in $S_4$ the subgroup $H=\langle (12)(34),(13) \rangle$, which is of order $8$. $H$ is clearly $1$-transitive since it contains the Klein $4$-group of $S_4$, which is transitive. Moreover, the transitivity of $H$ is not sharp because of its order. Finally, $H$ is not $2$-transitive, as the only proper $2$-transitive subgroup of $S_4$ is $A_4$.