The reduction of the structute group of $\mathbb{R}^n$ -fiber bundles to a special subgroup of $Homeo(\mathbb{R}^n)$

Let $ G$ be the group of all homeomorphisms $ f$ of $ \mathbb{R}^n$ which satisfy $ $ f(x+m)=f(x)+m,\quad \forall m\in \mathbb{Z}^n$ $

In the other words, $ G$ is the group of all equivariant homeomorphisms of Euclidean space with respect to the standard action of $ \mathbb{Z}^n$ on $ \mathbb{R}^n$ .

Is there a fiber bundle whose fibers are homeomorphic to$ \mathbb{R}^n$ but the structure group of the bundle can not be reduced to $ G$ ? Is there a manifold $ M$ such that the structure group of $ TM\to M$ , which is counted as a fiber bundle, can not be reduced to $ G$ ?