We compare the smooth and deformation equivalence of actions of
finite groups on $K3$ -surfaces by holomorphic and antiholomorphic
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 first that, 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, that is, reduces to
multiplication by $\pm 1$ .