I was wondering if there is an easy counter example to what follows:
Suppose that $E$ is contractible CWcomplex and $G_{1}, G_{2}$ are two isomorphic groups acting freely and continuously on $E$.
Is it true that the two actions are conjugated ?
I was wondering if there is an easy counter example to what follows: Suppose that $E$ is contractible CWcomplex and $G_{1}, G_{2}$ are two isomorphic groups acting freely and continuously on $E$. Is it true that the two actions are conjugated ? 


You can make a free group of rank two act on the plane (or the hyperbolic plane if you prefer) in two ways such that the orbit spaces are not homeomorphic: one is a punctured torus and the other is a three times punctured sphere. 


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. 

