abstract: I will present recent joint work with A. Della Vedova and Y. Shi concerning the persistence of csck metrics under (Galois) covers. We will show how the recent breakthrough by Cheng-Chen can be used to control the properness of the K-energy under covers, hence producing many new families of csck manifolds. Previous results by A-Ghigi-Pirola, Li-Sun and Cheng-Chen will be also discussed, and how the general theory of existence of Galois covers in the cyclic (classical), abelian (by Pardini) and non-abelian (by Catanese-Perroni and others) case can be applied.