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?