  Let $f:X\to Y$ be a finite ramified Galois covering of algebraic varieties
defined over the complex numbers. In this paper, we prove some structure
theorems for such coverings in the case that the non-abelian Galois group of
the cover is metabelian (in particular metacyclic). Our results extend the
previous ones, obtained by several authors, in the case of abelian and dihedral
Galois coverings. In analogy with the abelian (and generalizing the dihedral)
case, we find "building data" of metabelian covers from which the cover can be
reconstructed. In addition to analyzing several examples, we compute invariants
and eigensheaves of the group actions of the coverings under study with some
applications.
