  The automorphism group of the Galois covering induced by a pluri-canonical
generic covering of a projective space is investigated. It is shown that by
means of such coverings one obtains, in dimensions one and two, serieses of
specific actions of the symmetric groups $S_d$ on curves and surfaces not
deformable to an action of $S_d$ which is not the full automorphism group. As
an application, new DIF $\ne$ DEF examples for $G$-varieties in complex and
real geometry are given.
