A celebrated theorem of Hopf classifies round spheres in $\mathbb{R}^3$ as the unique spheres with constant mean curvature (CMC). Uniqueness of CMC spheres has been extended to other simply-connected homogeneous three-manifolds, provided that the dimension of the isometry group is 6 or 4. We will discuss this problem in the remaining case, when the isometry group of the ambient manifold is three-dimensional. These ambient geometries are always achieved by Lie groups equipped with a left invariant metric.