Let $G$ be a $n$-transitive permutation group of a finite set $X$. Let $\alpha\ne id_X\in G$ be permutation that fixes $k$ elements of $X$. What other condition is needed to ensure that there exists $\beta\in G$ such that the commutator of $\alpha,\beta$ is not equal to the identity? Is there a formulaic way to construct this $\beta$?
Edit:
I want to be able to mimic the construction of $\tau$ used in the proof of Lemma 2.4 on p.2 of the following document for $n$-transitive permutation groups: https://kconrad.math.uconn.edu/blurbs/grouptheory/Ansimple.pdf