  We study the dynamics of the spin system that consist of a positively charged
II-VI semiconductor quantum dot doped with a single Cr$^+$ ion. The resonant
photoluminescence (PL) of the positively charged exciton coupled with the
Cr$^+$ spin is used to analyze the main spin relaxation channels. The intensity
of the resonant PL is reduced by an optical pumping of the spin of the resident
hole-Cr$^+$ complex that can be seen as a nano-magnet. The spin memory can be
partially erased by a non-resonant optical excitation. This leads to an
increase of the resonant PL signal. The resonant PL is co-circularly polarized
and corresponds to relaxation channels that conserve the Cr$^+$ spin $\vert S_z
\vert$. The observation in the resonant-PL excitation spectra of optical
transitions with a change of the Cr$^+$ spin permits to determine the magnetic
anisotropy of the magnetic atom. Optical pumping, auto-correlation measurements
and the power dependence of the PL intensity distribution show that the
effective temperature of the hole-Cr$^+$ spin system is affected by the optical
excitation through the local generation of phonons.
