The classifying space of the 1+1 dimensional $G$-cobordism category

For a finite group $G$, we define the $G$-cobordism category in dimension two. We show there is a one-to-one correspondence between the connected components of its classifying space and the abelianization of $G$. Also, we find an isomorphism of its fundamental group onto the direct sum $\mathbb{Z}\oplus \Omega_2^{SO}(BG)$, where $\Omega_2^{SO}(BG)$ is the free oriented $G$-bordism group in dimension two, and we study the classifying space of some important subcategories. We obtain the classifying space has the homotopy type of the product $G/[G,G]\times S^1\times X^G$, where $\pi_1(X^G)=\Omega_2^{SO}(BG)$. Finally, we present some results about the classification of $G$-topological quantum field theories in dimension two.