Fundamental group of quotient disk

Question

Consider the disk $D^{2}$ in $\mathbb{R}^{2}$. By taking out two disjoint, smaller disks within $D^{2}$, we obtain a disk of genus 2. Now consider identifying the boundaries of the two deleted circles and the original boundary of the disk. How would one go about that one may perform such an identification in two distinct ways, such that the fundamental group of the resulting spaces are not isomorphic to each other?