  A complex irreducible character of a finite group G is said to be p-constant,
for some prime p dividing the order of G, if it takes constant value at the set
of p-singular elements of G. In this paper we classify irreducible p-constant
characters for finite reflection groups, nilpotent groups and complete monomial
groups. We also propose some conjectures about the structure of the groups
admitting such characters.
