= Kernel cover of a real projective plane wedged with a circle
For $C_2*\mathbb Z\to S_3$ taking the two generators to a transposition and a $3$-cycle, the kernel's six-sheeted regular <covering space> consists of three spheres and two circles. Each sphere has two marked points, and each circle meets one point on each sphere. It retracts onto a rank-four <graph> by replacing each sphere by a meridian arc, but admits no <deformation retraction> onto any <graph> because its degree-two <homology (mathematics)> is $\mathbb Z^3$.
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