We have studied the magnetotransport properties of a manganese (Mn)-modulation-doped high mobility two-dimensional hole system in a strained InAs quantum well (QW) structure. At precisely T=600 mK a phase transition from paramagnetism to ferromagnetism can be observed by a change of the low-field magnetotransport behavior and hysteresis. In the magnetically ordered phase a superposition of positive magnetoresistance and weak antilocalization was detected in the longitudinal resistance R(xx) and in the Hall resistance R(xy) a superposition of normal, anomalous, and planar Hall effects demonstrating spontaneous magnetization in the QW plane was detected. From extensive analysis of the temperature and magnetic field dependence of the Shubnikov-de Haas oscillations we deduce the effective mass, transport, and quantum-scattering times. The latter indicates presence of small-range scattering potential. From corrections to the Drude conductivity we determine the impurity interaction time, which is significantly reduced in the ferromagnetic phase indicating interaction of the two-dimensional free holes' spin with the localized magnetic moments of 5/2 from Mn ions.