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]: