In my algebraic topology course we briefly discussed automorphism groups of covering spaces before moving onto deformation retracts. In passing, my professor mentioned that it is easy to come up with covering spaces of $S^1 \vee S^1$ whose automorphism group is $\mathbb{Z}_3, \mathbb{Z}_6$, $D_4$, and $A_4$.
After some thought, I was able to come up with (i.e. draw covering spaces of $S_1 \vee S_1$ having automorphism groups isomorphic to $\mathbb{Z}_3, \mathbb{Z}_6$ and $D_4$. However, after several days of trying, I have been unable to give an example of a covering space of $S^1 \vee S^1$ whose automorphism is group is isomorphic to $A_4$.
How should I go about finding such an example? Any input is appreciated!