On transitive sets of derangements in primitive groups

We construct a primitive permutation action of the Steinberg triality group ${}^3{D}_4(2)$ of degree $4064256$ and show that there are distinct points $\alpha ,\beta$ such that there is no derangement $g\in {}^3{D}_4(2)$ with ${\alpha }^g=\beta$. This answers a question by John G. Thompson (Problem 8.75 in the Kourovka Notebook) in the negative.