Brauer characters with cyclotomic field of values

It has been shown in an earlier paper [G. Navarro, Pham Huu Tiep, Rational Brauer characters, Math. Ann. 335 (2006) 675–686] that, for any odd prime pp, every finite group of even order has a non-trivial rational-valued irreducible pp-Brauer character. For p=2p=2 this statement is no longer true. In this paper we determine the possible non-abelian composition factors of finite groups without non-trivial rational-valued irreducible 2-Brauer characters. We also prove that, if p≠qp≠q are primes, then any finite group of order divisible by qq has a non-trivial irreducible pp-Brauer character with values in the cyclotomic field Q(exp(2πi/q))Q(exp(2πi/q)).