Four-sheeted cover of a wedge of circles and a real projective plane

For $n\ge 1$, mapby sending the projective-plane generator and the first circle generator to the two standard generators. The kernel defines a connected regular four-sheeted cover ${\tilde{X}}_n$. Its lifted cellular boundary in degree two is multiplication by $1+v$ on $\mathbb{Z}[C_2\times C_2]$, so