= Four-sheeted cover of a wedge of circles and a real projective plane
For $n\geq1$, map
$$
\pi_1\left(\bigvee_{i=1}^nS^1\vee\mathbb{RP}^2\right)
\longrightarrow C_2\times C_2
$$
by sending the projective-plane generator and the first circle generator to the two standard generators. The kernel defines a connected regular four-sheeted cover $\widetilde X_n$. Its lifted cellular boundary in degree two is multiplication by $1+v$ on $\mathbb Z[C_2\times C_2]$, so
$$
H_2(\widetilde X_n;\mathbb Z)\cong\mathbb Z^2.
$$
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