Catalytically active macromolecules are envisioned as key building blocks in the development of artificial nanomotors. However, theory and experiments report conflicting findings regarding their dynamics. The lack of consensus is mostly caused by the limited understanding of the specifics of self-propulsion mechanisms at the nanoscale. Here, we study a generic model of a self-propelled nanoparticle that does not rely on a particular mechanism. Instead, its main assumption is the fundamental symmetry of microscopic dynamics of chemical reactions: the principle of microscopic reversibility. Significant consequences of this assumption arise if we subject the particle to the action of an external time-periodic force. The particle diffusion coefficient then becomes enhanced compared to the unbiased dynamics. The enhancement can be controlled by the force amplitude and frequency. We also derive the power spectrum of particle trajectories. Among the new effects stemming from the microscopic reversibility are the enhancement of the spectrum at all frequencies and sigmoid-shaped transitions and a peak at characteristic frequencies of rotational diffusion and external forcing. Microscopic reversibility is a generic property of a broad class of chemical reactions. Therefore, we expect that the presented results will motivate new experimental studies aimed at testing our predictions. This could provide new insights into the dynamics of catalytic macromolecules.