X-Factor started out as the original X-Men pretending to the world that they were mutant hunters, while secretly using the information provided by the public to help find young mutants.
Over the years, various incarnations of the group have arisen, with the common denominator that they either function as a group out of financial considerations, or that they pretend to do that.
Notes:
First appearance in X-Factor (Marvel, 1986 series) #1 (February 1986).