  We compare the smooth and deformation equivalence of actions of finite groups
on K3-surfaces by holomorphic and anti-holomorphic transformations. We prove
that the number of deformation classes is finite and, in a number of cases,
establish the expected coincidence of the two equivalence relations. More
precisely, in these cases we show that an action is determined by the induced
action in the homology. On the other hand, we construct two examples to show
that, first, in general the homological type of an action does not even
determine its topological type, and second, that K3-surfaces $X$ and $\bar X$
with the same Klein action do not need to be equivariantly deformation
equivalent even if the induced action on $H^{2,0}(X)$ is real, i.e., reduces to
multiplication by $\pm 1$.
