We show that all nontrivial finite subgroups of derived automorphisms of K3 surfaces of Picard number one have order two, and give explicit 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. Joint work with Kuan-Wen Lai.