Let C⊂P2 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 πP:C⇢P1 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⁎⊂P2⁎ which preserves the fibers of the projection from a certain point . In this article we improve this result, and we investigate the Galois group of . In particular, we study both when is a Galois point, and when deg(πP) is prime and . 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.