  Let $C \subset \mathbb{P}^2$ be a plane curve of degree at least three. A
point $P$ in projective plane is said to be Galois if the function field
extension induced by the projection $\pi_P: C \dashrightarrow \mathbb P^1$ from
$P$ is Galois. Further we say that a Galois point is extendable if any
birational transformation induced by the Galois group can be extended to a
linear transformation of the projective plane. This article is the second part
of [2], where we showed that the Galois group at an extendable Galois point $P$
has a natural action on the dual curve $C^* \subset \mathbb{P}^{2*}$ which
preserves the fibers of the projection $\pi_{\overline{P}}$ from a certain
point $\overline{P} \in \mathbb{P}^{2*}$. In this article we improve such a
result, and we investigate the Galois group of $\pi_{\overline{P}}$. In
particular, we study both when $\overline{P}$ is a Galois point, and when ${\rm
deg} \ (\pi_P)$ is prime and ${\rm deg} \ (\pi_{\overline{P}}) = 2{\rm deg} \
(\pi_P)$. As an application, we determine the number of points at which the
Galois groups are certain fixed groups for the dual curve of a cubic curve.
