  We present a spectroscopic analysis of a 1-year intensive monitoring campaign
of the 22-Myr old planet-hosting M dwarf AU Mic using the HARPS spectrograph.
In a companion paper, we reported detections of the planet radial velocity (RV)
signatures of the two close-in transiting planets of the system, with
respective semi-amplitudes of 5.8 $\pm$ 2.5 m/s and 8.5 $\pm$ 2.5 m/s for AU
Mic b and AU Mic c. Here, we perform an independent measurement of the RV
semi-amplitude of AU Mic c using Doppler imaging to simultaneously model the
activity-induced distortions and the planet-induced shifts in the line
profiles. The resulting semi-amplitude of 13.3 $\pm$ 4.1 m/s for AU Mic c
reinforces the idea that the planet features a surprisingly large inner
density, in tension with current standard models of core accretion. Our
brightness maps feature significantly higher spot coverage and lower level of
differential rotation than the brightness maps obtained in late 2019 with the
SPIRou spectropolarimeter, suggesting that the stellar magnetic activity has
evolved dramatically over a $\sim$1-yr time span. Additionally, we report a
3-$\sigma$ detection of a modulation at 8.33 $\pm$ 0.04 d of the He I D3
(587.562 nm) emission flux, close to the 8.46-d orbital period of AU Mic b. The
power of this emission (a few 10$^{17}$ W) is consistent with 3D
magnetohydrodynamical simulations of the interaction between stellar wind and
the close-in planet if the latter hosts a magnetic field of $\sim$10 G.
Spectropolarimetric observations of the star are needed to firmly elucidate the
origin of the observed chromospheric variability.
