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For a finite group $G$, we define the $G$-cobordism category in dimensiontwo. We show there is a one-to-one correspondence between the connectedcomponents of its classifying space and the abelianization of $G$. Also, wefind an isomorphism of its fundamental group onto the direct sum$\mathbb{Z}\oplus \Omega_2^{SO}(BG)$, where $\Omega_2^{SO}(BG)$ is the freeoriented $G$-bordism group in dimension two, and we study the classifying spaceof some important subcategories. We obtain the classifying space has thehomotopy 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 theclassification of $G$-topological quantum field theories in dimension two.

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