I was thinking about the question Is a HNN extension of a virtually torsion-free group virtually torsion-free? and I started wondering whether or not $G = \mathbb{Z}\times C_2$ was a counterexample to the question. In the process, I came across the following question:
Is $\mathbb{Z}* C_2=\left< t,x|x^2=1 \right>$ virtually torsion-free?
A group is said to be virtually torsion-free if it has a finite index subgroup which does not contain elements of finite order.
I suspect that the answer is no because words of the form $t^nxt^{-n}$ all have order two within $\mathbb{Z}*C_2$. If $\mathbb{Z}*C_2$ has a finite index torsion-free subgroup $T$ then the cosets $t^nxt^{-n}T$ cannot all be distinct.