I am interested in actions of a finite group $G$ on a finite set $X$ which are $k$- but not $(k+1)$-transitive for $6\leq k\leq |X|-1$. In "Permutation Groups" (by Dixon, Mortimer, 1996) it is said in section 7.3 that
Except for the alternating and symmetric groups, the only finite groups which are 4- or 5-transitive are the Mathieu groups...
And similarly in "Finite Permutation Groups and Finite Simple Groups" (Bull. London Math. Soc., by Cameron, 1981) that
The symmetric group of degree $n$ is $n$-transitive, and the alternating group is $(n-2)$-transitive. Apart from these examples, no known finite group is 6-transitive...
These formulations remain a bit vague for me, though. I would like to know if here "alternating/symmetric group" means "alternating/symmetric group in its natural action" (i.e. $A_n$ or $S_n$ acting on $\{1,...,n\}$ by applying the permutations).
More concretely, might there be an alternating/symmetric group with some action - different from the natural action - satisfying my situation of interest? Or does the literature mean that, once an alternating/symmetric group has a $k$-transitive action with $k\geq 6$, this must be the natural action?