= Solution
Use two sheet points $v_0,v_1$. Attach a copy of $\mathbb{RP}^2$ at each, each mapping identically onto the projective-plane summand. Put an oriented $x$ edge from $v_0$ to $v_1$ and another from $v_1$ to $v_0$, each mapping once along the cut-open <circle>. This is a <connected> two-sheeted cover.
Its underlying two-edge <circle> contributes one infinite cyclic free factor, and its two attached projective planes contribute two factors of order two. Collapsing one of the connecting edges, or applying <Seifert-van Kampen theorem>, gives
$$
\pi_1(\widetilde X,v_0)\cong\mathbb Z*C_2*C_2.
$$
Choose the first $x$ edge as a tree path to $v_1$. The <circle> loop projects to $x^2$, and the two projective-plane generators project to $y$ and $xyx^{-1}$. By injectivity from the preceding proof,
$$
\boxed{\langle x^2,\ y,\ xyx^{-1}\rangle\cong\mathbb Z*C_2*C_2}.
$$
The order-two relations come from the respective projective planes; there are no additional relations between these free factors.
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