Automorphisms of Galois coverings of generic $m$-canonical projections

We investigate the automorphism groups of Galois coverings induced by pluricanonical generic coverings of projective spaces. In dimensions one and two, it is shown that such coverings yield sequences of examples where specific actions of the symmetric group $S_d$ on curves and surfaces cannot be deformed together with the action of $S_d$ into manifolds whose automorphism group does not coincide with $S_d$. As an application, we give new examples of complex and real $G$-varieties which are diffeomorphic but not deformation equivalent.