Let $X$ be the following figure:
I want to find the fundamental groups $\pi_{1}(X)$ and $\pi_{1}(X/\beta)$ (up to isomorphism), determine whether $B$, the part of the image including the "$\beta$-circle" and below, is a retract of $X$, and determine whether the equator cirlce (the "filling" of the circle contains the equator) is a retract of $X$.
Unfortunately, I wasn't able to get too far on my own, but here's everything I got: first, $X$ is homotopic to a sphere with a "filled-in" equator, because we can contract the "filling" into a single disc. Then, we can contract that disc into a point, and get the wedge sum of two spheres, of which one has a disc-shaped hole with the $\alpha$ identification, and the other one has a circle with a triple $\beta$ identification. I've included a picture in case my explanation isn't clear: 
I know the upper sphere of the wedge sum is the real projective plane, whose fundamental group is $\mathbb{Z_{2}}$; however, I can't find the other one. $\pi_{1}(X)$ would then be the free product of $\mathbb{Z_{2}}$ and the other fundamental group.
As far as $\pi_{1}(X/\beta)$ goes, I would hazard to guess that we could then contract the circle on the first picture with the $\beta$ identification, and that $\pi_{1}(X/\beta) \cong \mathbb{Z_{2}}$ then. Is this true? It sounds too easy to be true, but that's the only reasoning that I could think of.
As far as the retract questions go, I have made no progress in answering them. I would appreciate any assistance with this problem.