  Given a flat, finite group scheme G finitely presented over a base scheme we
introduce the notion of ramified Galois cover of group G (or simply G-cover),
which generalizes the notion of G-torsor. We study the stack of G-covers,
denoted with G-Cov, mainly in the abelian case, precisely when G is a finite
diagonalizable group scheme over Z. In this case we prove that G-Cov is
connected, but it is irreducible or smooth only in few finitely many cases. On
the other hand, it contains a 'special' irreducible component Z_G, which is the
closure of BG and this reflects the deep connection we establish between G-Cov
and the equivariant Hilbert schemes. We introduce 'parametrization' maps from
smooth stacks, whose objects are collections of invertible sheaves with
additional data, to Z_G and we establish sufficient conditions for a G-cover in
order to be obtained (uniquely) through those constructions. Moreover a toric
description of the smooth locus of Z_G is provided.
