Here's a question I'm working on for my Group Theory class:

Suppose $\displaystyle G|X$ is a 3-transitive permutation group. Suppose further that $\displaystyle |X|=n$ is NOT a power of 2, and that $\displaystyle G\ncong S_n$.

Prove or disprove: if $\displaystyle N$ is a proper normal subgroup of $\displaystyle G$, then the action $\displaystyle N|X$ is 2-transitive.

I'm pretty sure the statement is true (taking "proper" to mean $\displaystyle N\neq {e},G$). I'm not really sure where to start though.