  We prove that all nontrivial finite subgroups of derived automorphisms of K3
surfaces of Picard number one have order two and give formulas for the numbers
of their conjugacy classes. We also obtain a similar result for the subgroups
which are finite up to shifts. This in turn shows that such a K3 surface admits
an associated cubic fourfold if and only if it has a derived automorphism of
order three up to shifts. These results are achieved by proving that such a
subgroup fixes a Bridgeland stability condition up to $\mathbb{C}$-actions. We
also establish similar existence results for curves, twisted abelian surfaces,
generic twisted K3 surfaces, and standard autoequivalences on surfaces.
