I am representing $S^1 \vee S^1$ via the following diagram:
I have the following connected covering of $S^1 \vee S^1$, call it $p: \tilde{X} \rightarrow X$, for $X=S^1 \vee S^1$.
I am having difficulty in computing $p_{*}(\pi_1(\tilde{X}, \tilde{x}_0))$, where $p_{*}$ indicates the image of the map $p$ under the $\pi_1$ functor.
I know that $\pi_1(\tilde{X},\tilde{x}_0) \cong <\alpha, \beta, \gamma, \sigma, \theta, \mu>$ , but my issue lies in finding the images $\alpha, \dots, \mu$ under $p_*$. My naive guess would be that $\alpha = b, \beta = a^2, \gamma = b^2, \sigma = a^2, \theta = b^2, \mu = a$ but I believe this doesn't respect the basepoint $\tilde{x}_0$.
The concern you mentioned at the end of your post is pretty accurate. It may be helpful to recall how you arrived at the fundamental group of your covering space, this should give you an answer to your question. The computation can be done via the Seifert-Van-Kampten theorem, that inductively gives you the fundamental group your wedge sum. But to make life easier you will change the basepoint at each step, which is possible for path connected spaces. The change of basepoint also has an explicit isomorphism of fundamental groups by picking a path and conjugating your loop by it (so of the form $f \mapsto sfs^{-1}$). For you that means remembering the path you had to chose to get to the loop you want, I think this is the idea you had, that each loop represents a generator. So $p(\alpha )= b^{-1}a^{-1}bab$, $p(\beta) = b^{-1}aab$ ... . Algebraic Topology by A.Hatcher also gives those Covering spaces as an example.
Edit: forgot to take inverses