Let $\Gamma$ be the semidirect product of $\mathbb{Z}$ and $\mathbb{Z}/4$,
where the action of $\mathbb{Z}/4$ on $\mathbb{Z}$ is defined by $\bar{k} \cdot x = (-1)^k x$. Clearly $\Gamma$ has virtual cohomological dimension (*vcd*) one.

Is it possible to construct a one-dimensional contractible CW complex $X$ such that $\Gamma$ acts cellularly on $X$ and the cells have only finite stabilizers?

$\mathbb{Z}$ acts freely on $\mathbb{R}$ by right-shift. I tried without success to adjust this action to obtain an action of $\Gamma$. So any help is appreciated.

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