D4-type subgroups of F4(q)

In this paper we treat the action of the simple group $F_4(q)$ on the cosets of subgroups $D_4(q)$, ${}^2 \nts D_4(q)$ and ${}^3 \nts D_4(q)$ and their extensions by graph automorphisms. We obtain the ranks and decompose the corresponding permutation characters; we show that, even allowing for the application of field automorphisms, the only two primitive multiplicity-free actions arising are those of $F_4(2)$ on cosets of $D_4(2).S_3$ and ${}^3 \nts D_4(2).3$. For these two actions, we calculate the subdegrees; we find that all suborbits are self-paired, but that the action gives rise to no distance-transitive graph.