Monolithic characters of real groups

In this article, we are concerned with the nonlinear primitive and monolithic characters of a finite real group. We give some criteria for determining whether or not a finite group is real by using its monolithic characters. Also, we classify all finite real groups which have at most two nonlinear monolithic characters and obtain a result for real groups with three nonlinear monolithic characters. Finally, we prove that there are not any nonlinear primitive irreducible characters of a finite solvable real group having at most five nonlinear monolithic characters.