= Solution
Take a <CW complex> with one vertex, loops labelled $a,b$, and one two-cell attached along $a^2b^3a^3b^2$. The <Seifert-van Kampen theorem> gives the required <fundamental group>. Connected double <covering spaces> correspond to index-two subgroups, equivalently kernels of surjective homomorphisms $G\to C_2$. The relation imposes $5a+5b=0$ modulo two, so the unique nonzero possibility sends both $a,b$ to one. Thus \b[there is a unique connected double cover up to covering isomorphism.]
Construct it with two vertices; both labelled edges switch sheets, and attach the two lifts of the two-cell. Contract one of the $a$ edges to a spanning tree. With coset representatives $1,a$, the remaining <Schreier generators> can be taken as $A=a^2$, $B=ba^{-1}$ and $C=ab$. Reading the two lifts of the attaching word, starting on the two respective sheets, gives $ABCBA^2BC$ and $ACBCACB$. Thus
$$
\boxed{\pi_1(Y)=\langle A,B,C\mid ABCBA^2BC=1,\ ACBCACB=1\rangle}.
$$
For example, a letter $a$ contributes nothing on the first sheet and $A$ on the second; a letter $b$ contributes $B$ on the first and $C$ on the second. Both letters then switch sheets, which directly verifies the two relators.
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