Describing units of integral group rings up to commensurability

We restrict the types of 2×22×2-matrix rings which can occur as simple components in the Wedderburn decomposition of the rational group algebra of a finite group. This results in a description up to commensurability of the group of units of the integral group ring ZGZG for all finite groups G that do not have a non-commutative Frobenius complement as a quotient.