  This paper continues discussions in the author's previous paper about the
Misiurewicz polynomials defined for a family of degree $d \ge 2$ rational maps
with an automorphism group containing the cyclic group of order $d$. In
particular, we extend the sufficient conditions that the Misiurewicz
polynomials are irreducible over $\mathbb{Q}$. We also prove that the
Misiurewicz polynomials always have an irreducible factor of large degree.
