  Let $M$ be a closed orientable 3-manifold with a genus two Heegaard splitting
$(V_1, V_2; F)$ and a non-trivial JSJ-decomposition, where all components of
the intersection of the JSJ-tori and $V_i$ are not $\partial$-parallel in $V_i$
for $i=1,2$. If $G$ is a finite group of orientation-preserving diffeomorphisms
acting on $M$ which preserves each handlebody of the Heegaard splitting and
each piece of the JSJ-decomposition of $M$, then $G\cong \mathbb{Z}_2$ or
$\mathbb{D}_2$ if $V_j\cap(\cup T_i)$ consists of at most two disks or at most
two annuli.
