In earlier work [Math. Ann. 335, No. 3, 675-686 (2006; Zbl 1106.20006)], the first two authors proved that for any odd prime \(p\), every finite group of even order has a non-trivial rational-valued irreducible \(p\)-Brauer character. For \(p=2\) it is known that this is false for all groups \(L_2(3^{2f+1})\) (\(f=1,2,\dots\)). The main results of the paper are the following: Theorem A. Let \(G\) be a finite group with no nontrivial rational irreducible 2-Brauer character. Then every nonabelian composition factor of \(G\) is isomorphic to \(L_2(3^{2f+1})\) for some \(f\). Theorem B. Let \(p\) and \(q\) be distinct primes. Suppose that \(G\) is a finite group of order divisible by \(q\). Then \(G\) has a nontrivial irreducible \(p\)-Brauer character with values in the cyclotomic field of \(q\)-th roots of unity. The proofs of these results depend on CFSG.