Enriques surfaces

Unlike this is the case for K3-surfaces, there may exist automorphisms which act trivially in cohomology. This was first observed by Lieberman (not published). He found an Enriques surface with an involution which acts trivially on the rational cohomology group. This involution turned out not be the identity on integral cohomology (there is torsion in cohomology: the canonical class gives a non-trivial 2-torsion element). Barth and Peters gave an example of an involution which does act trivially in integral cohomology.
It turns out (Mukai, oral communication) that this is the only possible example.
Apart from Lieberman's example there are a few other possible examples exhibiting a similar phenomenon.
Mukai and Namikawa started this study which has been completed recently by Mukai. See [M1], [M2]:

[M1] S. Mukai: Numerical trivial involutions on Enriques surfaces http://www.kurims.kyoto-u.ac.jp/preprint/file/RIMS1544.pdf
[M2] S. Mukai: Kummer's quartics and numerically reflexive involutions of Enriques surfaces http://www.kurims.kyoto-u.ac.jp/preprint/file/RIMS1633.pdf