  Motivated by the work of Birman about the relationship between mapping class
groups and braid groups, we discuss the relationship between the orbit braid
group and the equivariant mapping class group on the closed surface $M$ with a
free and proper group action in this article. Our construction is based on the
exact sequence given by the fibration $\mathcal{F}_{0}^{G}M \rightarrow
F(M/G,n)$. The conclusion is closely connected with the braid group of the
quotient space. Comparing with the situation without the group action, there is
a big difference when the quotient space is $\mathbb{T}^{2}$.
