Let a $3$-dimensional subspace $V$ of $\mathbb{R}^4$ be $$V=\{(x\_1,x\_2,x\_3,x\_4)\in\mathbb{R}^4\mid\sum\_{i=1}^4x\_i=0\}.$$ The alternating group $A\_4$ acts on $V$ by 
$$\sigma(x\_1,x\_2,x\_3,x\_4)=(x\_{\sigma(1)},x\_{\sigma(2)},x\_{\sigma(3)} ,x\_{\sigma(4)})$$ for any $\sigma\in A\_4$.


Since $V$ is linearly isometric to $\mathbb{R}^3$, $SO(3)$ acts on $V$ canonically. Moreover, $A\_4$ acts on $V$ as a subgroup of $SO(3)$. Hence we have a covering map
$$
A\_4\to SO(3)\to SO(3)/A\_4.
$$
Letting $A\_4$ act on $\mathbb{R}^4$ by permuting basis and attaching $\mathbb{R}^4$ as fibres, we have a vector bundle associated to the covering map
$$
\xi: \mathbb{R}^4\to SO(3)\times \_{A\_4}\mathbb{R}^4\to SO(3)/A\_4.
$$


**Question:** (1). Is $\xi$ a trivial vector bundle? 


(2). Is $\xi^{\oplus 2}$ (2-fold Whitney sum) a trivial vector bundle?


(3). What is the smallest integer $n$ such that $\xi^{\oplus n}$ is a trivial vector bundle?


My attempt: I plan to compute the Stiefel-Whitney class. But I am not sure whether all Stiefel-Whitney classes of $\xi$ are trivial or not?



triviality of Whitney sums of a vector bundle
There is a simple general argument showing that $\xi$ is trivial: Take any homogeneous space $G/H$ and any representation $V$ of $G$ and restrict the representation to $H$. Then the homogeneous vector bundle $G\times\_H V\to G/P$ is a trivial as a vector bundle. A trivialization can be written down explicitly. It is induced by the map $G\times V\to (G/H)\times V$ defined by $(g,v)\mapsto (gH,g\cdot v)$. This is evidently $H$-invariant and factorizes to an isomorphism $G\times\_H V\to (G/H)\times V$ of vector bundles. 


Edit (following the comments by @Sebastian\_Goette and @Asghar\_Ghorbanpour): This argument can be applied both to the three dimensional representation of $A\_4$ called $V$ in the question and to the representation on $\mathbb R^4$ by permutation of coordinates. The latter representation also extends to $SO(3)$ as the direct sum of the representation on $V$ and a trivial representation. 


