We present a detailed numerical analysis of the magnetization auto-oscillations induced in a thin NiFe nanowire by a direct spin polarized current injected via a square-shaped CoFe nanomagnet (a system experimentally studied in C. Boone et al., Phys. Rev. B 79, 140404 (2009)). We demonstrate that all auto-oscillation modes in the nanowire are localized under the nanocontact for magnetic field applied in the plane of the nanowire. This mode localization is induced by a strong stray magnetic field acting on the NiFe nanowire from the CoFe current injector. We find that the auto-oscillation frequency, power and the frequency shift with the current strongly depend on the exchange constant of NiFe. We also find that the auto-oscillation power depends non-monotonically on the CoFe saturation magnetization, and demonstrate that this effect has its origin in resonant excitation of the CoFe eigenmodes by magnetization oscillations in the NiFe nanowire. The calculated dependence of the oscillation frequency on current is in a good agreement with the experiment. However, the agreement between theory and experiment for the oscillation power is unsatisfactory. Finally, we have shown that an auto-oscillatory mode propagating along the nanowire in the system under study is possible when a sufficiently strong out-of-plane field is applied.