Dixon's Book:
Exercise 2.1.6: Suppose $G$ is $k$-transitive for some $k > 2$, and $N$ is a nontrivial normal subgroup of G. Show that N is $(k-1)$-transitive.
But we have in Wielandt's book:
Theorem 9.9. A normal subgroup $Ν \neq 1$ of a $k$-fold transitive group $G \neq S_n$ of degree n $(k > 2)$ is in general $(k-1)$-fold transitive. There are exceptions only in the case $n=2^m$, $k = 3$, in which case $N$ may be a regular elementary Abelian $2$-group.
Exceptions in the exercise does not mentioned!