We explore the mixing behavior of random walks on Cayley graphs of non-Abelian groups, focusing on nilpotent and dihedral groups, both of which can be viewed as “almost Abelian.” For nilpotent groups, we study both walks driven by k randomly chosen generators and conjugacy-invariant walks, showing in different settings how their mixing is governed by the Abelianization. In particular, for k random generators, cutoff occurs with high probability throughout the regime 1 ≪ log k ≪ log |G|, at the entropic time of the projected walk. For dihedral groups, the cutoff time behaves differently from the entropic time on the cyclic subgroup. A common perspective behind these results comes from analyzing the entropy evolution of suitable auxiliary processes, reflecting the underlying commutativity structure of the group. This talk is based on several works, including joint work with Jonathan Hermon and Renyu Rao.