I am trying to solve this question from Lee's Topological Manifolds but I am not sure how to compute the covering group/automorphism group.
What I have been thinking of doing is using the following theorem. This would require me to find the induced homomorphism of the fundamental group by the covering space. However, I'm not sure how to exactly do this. How can I go about computing the induced homomorphism by q so that I can apply this theorem?
I know the fundamental group of the covering space should be the free group on three generators since it is the bouquet of three circles. Then how can I compute $q_*\pi_1(\tilde X, e)$?
Here is the problem I am attempting to solve:
