Describing units of integral group rings up to commensurability

We restrict the type of $2\times 2$-matrices 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 $\mathbb{Z}G$ for all finite groups $G$ that do not have a non-commutative Frobenius complement as a quotient.