In this paper we let Gbe the finite simple group F4(q), for qa prime power; we take Pto be the maximal parabolic subgroup with Levi subgroup having derived group B3(q), and Hto be the subgroup D4(q)generated by the long root elements of G. We consider the collection of (P, H)-double cosets in G; we obtain coset representatives and give the decomposition explicitly. As a consequence we are able to deduce that if q>2then the action of F4(q)on the cosets of the maximal subgroup D4(q).S3is not multiplicity-free, even if field automorphisms are applied.