TITLE: free action on contractible spaces
QUESTION [5 upvotes]: I was wondering if there is an easy counter example to what follows:
Suppose that $E$ is contractible CW-complex and $G_{1}, G_{2}$ are two isomorphic groups acting freely and continuously on $E$. 
Is it true that the two actions are conjugated ?

REPLY [6 votes]: The example I gave in comments has $E/G_1$ and $E/G_2$ compact (but not manifolds), at least if you use compact "blobs".
More explicitly, let $E$ be the subset of $\mathbb{R}^2$ given by the union of the $x$-axis and closed discs of equal radius around $(n,0)$ for every integer $n$. Let $G_1=\mathbb{Z}$ acting by integer translations, and $G_2=\mathbb{Z}$ acting by even integer translations.