The problem of transforming a locally asymptotically stabilizing time-varying control law to a globally stabilizing one with accelerated finite/fixed-time convergence is studied. The solution is based on an extension of the theory of homogeneous systems to the setting where the symmetry and stability properties only hold with respect to a part of the state variables. The proposed control design advances the kind of approaches first studied in [1], and relies on the implicit Lyapunov function framework. Examples of finite-time and nearly fixed-time stabilization of a nonholonomic integrator are reported.