  This is a continuation of our previous work [13]. Let $(\Sigma,g)$ be a
closed Riemann surface, where the metric $g$ has conical singularities at
finite points. Suppose $\mathbf{G}$ is a group whose elements are isometries
acting on $(\Sigma,g)$. Trudinger-Moser inequalities involving $\mathbf{G}$ are
established via the method of blow-up analysis, and the corresponding extremals
are also obtained. This extends previous results of Chen [7], Iula-Manicini
[21], and the authors [13].
