If you want to enumerate the finite-order automorphisms (up to conjugacy) I suggest the following exercise. The associated 3-manifold is Seifert fibred. So determine how the genus 2 surface is sitting in the Seifert manifold (horizontal incompressible surface).
This will give you a formula relating the various branch points of the monodromy to the Seifert data. Moreover, you should be able to go back-and-forth between the description of the Seifert-fibred space (unnormalized Seifert data, fibred over a genus 0, 1 or 2 surface) and the monodromy of the surface. So the classification of Seifert-fibred spaces basically gives you a dictionary for walking-through the finite-order automorphisms of a mapping class group.
