How do we classify the (path)connected covering spaces of $S^1 \lor \mathbb{R} \mathbb{P}^2$?
Of course, we start by classifying the subgroups of $\pi_1(S^1 \lor \mathbb{R} \mathbb{P}^2) = \mathbb{Z} * \mathbb{Z}_2$. However, I do not know how to. In this question: Covering spaces of $S^1\vee \Bbb RP^2$, the OP asks for the covering spaces corresponding to certain subgroups of $\mathbb{Z} * \mathbb{Z}_2$, but are those all subgroups of $\mathbb{Z} * \mathbb{Z}_2 = \langle a, b \ \mid \ b^2 = 1 \rangle$?
I think there are more subgroups, at least the ones of the form $\langle a^k \rangle$ and $\langle (ab)^k \rangle$ for an integer $k$, but are those all of the subgroups? And how would the covering spaces corresponding to these subgroups look like?