Double-cover of a Klein-bottle-esque Space

Let $G = \langle r,s | r^2 s^3 r^3 s^2 = 1\rangle$, the fundamental group of $A$.

By the Galois correspondence, the fundamental group of $B$ should be isomorphic to an index-two subgroup $H$ of $G$. Any subgroup of index two is normal, so $H$ is the kernel of some surjective homomorphism $\varphi\colon G \to \mathbb{Z}/2\mathbb{Z}$. From the relation $r^2 s^3 r^3 s^2 =1$, we can see that $5 \varphi(r) + 5 \varphi(s) = 0$ in $\mathbb{Z}/2\mathbb{Z}$, so the only possibility is $\varphi(r) = \varphi(s) = 1$. We conclude that neither $r$ nor $s$ lifts to a loop in the cover.

So the 1-skeleton of the double cover has two vertices and four edges, like this:

The single 2-cell in $A$ lifts to a pair of 2-cells in $B$, which are attached as follows:

You can now use this cell complex to get a presentation for the fundamental group of $B$.