  Let w be a group-word. Suppose that the set of all w-values in a profinite
group G is contained in a union of countably many cosets of subgroups. We are
concerned with the question to what extent the structure of the verbal subgroup
w(G) depends on the properties of the subgroups. We prove the following
theorem. Let C be a class of groups closed under taking subgroups, quotients,
and such that in any group the product of finitely many normal C-subgroups is
again a C-subgroup. If w is a multilinear commutator and G is a profinite group
such that the set of all w-values is contained in a union of countably many
cosets g_iG_i where each G_i is in C, then the verbal subgroup w(G) is
virtually-C. This strengthens several known results.
