It is known that every subgroup of the Mathieu group M23, one of the sporadic finite simple groups, appears as a subgroup of automorphisms of a K3 surface. This result was established by Mukai in 1988, and it was extended to the setting of derived automorphisms by Huybrechts in 2016. Conversely, when given a specific K3 surface, it is generally unknown which finite subgroups can possibly appear in its derived automorphism group. One of the goals of this talk is to present an answer to this question for K3 surfaces of Picard number one in terms of a certain family of Shimura curves. This talk is intended for a general audience in mathematics.