This is the second half of exercise 1.3.21 from Hatcher.
Let $Y$ be the space obtained by attaching a Möbius band $M$ to $\mathbb{R}P^2$ via a homeomorphism from its boundary circle to a circle in $\mathbb{R}P^2$ lifting to the equator in the covering space $S^2$ of $\mathbb{R}P^2$. Compute $\pi_1(Y)$, describe its universal cover, and describe the action of $\pi_1(Y)$ on the universal cover.
I've done the first half of the exercise, but this half has me a little mixed up. Mostly I'm just looking for a hint or a nudge in the right direction. Would the cell structure be something like: one $0$-cell, two $1$-cells, and two $2$-cells?
Thanks