Handlebody-preserving Finite Group Actions on Haken Manifolds with Heegaard Genus Two

Let M be a closed orientable 3-manifold of Heegaard genus two with a non-trivial JSJ-decomposition and G be a finite group of orientation-preserving smooth actions on M which preserves each handlebody of Heegaard splitting and each piece of the JSJ-decomposition of M . We will call some specific conditions of the Heegaard splitting and the JSJ-tori the condition A, . . . , and the condition F. In this article, we first determine the possible isomorphism types of G when the Heegaard splitting and the JSJ-tori satisfy the condition A, the condition B, . . . , or the condition F by Theorem 1.1 and Theorem 1.2. By these theorems and T. Kobayashi’s results, we determine the possible isomorphism types of G when M has separating JSJ-tori or the Heegaard splitting is strongly irreducible in the Main Theorem.