In this paper we begin by examining the action ofE6(q) on the cosets of the subgroupF4(q): we give the rank and subdegrees, and show that it is multiplicity‐free, that is, the constituents of the permutation character are all distinct. It is found that the suborbits correspond to conjugacy classes ofA2(q); we seek to explain this using the concept of ‘folding actions’. This enables the related action of2E6(q2onF4(q) to be treated with little extra effort.