The problem is to find a group $G$ and action of it on $\mathbb{R}^2$ such that $\mathbb{R}^2 / G \approx M \backslash \partial M$. ($M$ - Möbius strip)
I took $G = \mathbb{Z}$ and action of it defined by $r \cdot (x, y) = (x + r, (-1)^r y)$. The problem is that i can't show that this action induces required homemorphism. My idea was to consider some $A \subset R^2$ such that $A / \sim$, where $\sim$ is inherited equivalence relation induced by given action, is homeomorphic to $M \backslash \partial M$. For example, we can take $A = [-\frac{1}{2}, \frac{1}{2}] \times \mathbb{R}$, but to show that $\mathbb{R}^2/ G \approx A / \sim$ is where i got stuck. Any ideas how to proceed?