  The existence or non-existence of Einstein metrics on 4-manifolds with
non-trivial fundamental group and the relation with the underlying differential
structure are analyzed. For most points $(n,m)$ in a large region of the
integer lattice, the manifold $n\mathbb C \mathbb P^2\#m \overline{\mathbb C
\mathbb P^2}$ is shown to admit infinitely many inequivalent free actions of
finite cyclic groups and there are no Einstein metrics which are invariant
under any of these actions. The main tools are Seiberg-Witten theory, cyclic
branched coverings of complex surfaces and symplectic surgeries.
