An integral group ring \(\mathbb{Z} G\) is called coherent if finitely presented \(\mathbb{Z} G\)-modules and homomorphisms form an Abelian category. The aim of the paper is to prove that the integral group ring of a group containing a direct product of free groups is incoherent. In order to achieve this aim coherent augmented rings are characterized by means of derived modules, and it is proved that the integral group ring of the direct product of two free groups satisfies, in fact, a stronger property than incoherence, called the infinite kernel property: there is an exact sequence \( 0\to P\to\mathbb{Z} G^b\to\mathbb{Z} G^a\), where \(a\) and \(b\) are positive integers and \(P\) is a projective \(\mathbb{Z} G\)-module of infinite rank.