Let $V$ be a simple VOA of CFT-type satisfying $V'\cong V$ and $\sigma$ a
finite automorphism of $V$. We prove that if all $V$-modules are completely
reducible and a fixed point subVOA $V^\sigma$ is $C_2$-cofinite, then all
$V^\sigma$-modules are completely reducible and every simple
$V^{\sigma}$-module appears in some twisted or ordinary $V$-modules as a
$V^{\sigma}$-submodule. We also prove that $V_L^{\sigma}$ is $C_2$-cofinite for
any lattice VOA $V_L$ and $\sigma\in \Aut(V_L)$ lifted from any triality
automorphism of $L$.