One of the basic examples of a group action on a topological space is the $\mathbb{Z}\times\mathbb{Z}-$action on $\mathbb{R}^2$ whose quotient is a torus.
There is also an example of a $G-$action on $\mathbb{R}^2$ whose quotient is a Klein bottle, but $G\neq\mathbb{Z}\times\mathbb{Z}$, being instead the fundamental group of the Klein bottle.
My question is: is there a $\mathbb{Z}\times\mathbb{Z}-$action on $\mathbb{R}^2$ whose quotient is a Klein bottle?
I understand that, if there is such an action, it can't be properly discontinuous, otherwise the Klein bottle would have fundamental group $\mathbb{Z}\times\mathbb{Z}$.