Abstract: \mathbbZ_3-QCD is a QCD-like theory with strict center symmetry. We use the Polyakov-loop extended quark meson model (PQM) as a low-energy effective theory of \mathbbZ_3-QCD to study the RW transitions in different center symmetry breaking patterns. The flavor-dependent imaginary chemical potentials, namely (\mu_\rmu,\,\mu_\rmd,\,\mu_\rms)= \rmiT(\theta-2C\pi/3,\theta,\theta+2C\pi/3) are adopted, which guarantees the RW periodicity. The traditional and quark improved Polyakov-loop potentials are used, respectively. For N_\rmf\!=\!3 with C\!\ne\!1, the RW transition occurs at \theta\!=\!\pi/3 (mod 2\pi/3), which gets stronger when C declines from one to zero. When C\!=\!1, the RW transition happens at \theta\!=\!2\pi/3 (mod 2\pi/3) for N_\rmf\!=\!2+1, but \theta\!=\!\pi/3 (mod 2\pi/3) for N_\rmf\!=\!1+2. We find that all RW transition endpoints are triple points when C\!=\!1. We confirm that the RW transition becomes weaker and the deconfinement temperature gets lower when taking into account the quark back-reaction effect. However, the modification of the gluon sector due to the quark effect does not change the main conclusions mentioned above.