I am trying to understand isomorphism classes of bundles $$\mathbb{S}^1\hookrightarrow E\to \mathbb{RP}^2.$$ These are classified by homotopy classes $[\mathbb{RP}^2,BAut(\mathbb{S}^1)]$.
ATTEMPT 1. $Aut(\mathbb{S}^1)$ are the diffeomorphisms $\mathbb{S}^1\to \mathbb{S}^1$, and it is homotopically equivalent to $O(2).$ Therefore the universal classifying space $BAut(\mathbb{S}^1)$ is homotopically equivalent to $BO(2)$. Thus we are interested in homotopy classes in$[\mathbb{RP}^2,BO(2)]$.
Now topologically $O(2) \simeq SO(2) \sqcup SO(2)\simeq \mathbb{S}^1 \sqcup \mathbb{S}^1,$ (it has two connected component that are diffeomorphic to $\mathbb{S}^1$). Therefore $$BO(2)\simeq B\mathbb{S}^1\sqcup B\mathbb{S}^1 \simeq \mathbb{CP}^\infty \sqcup \mathbb{CP}^\infty.$$
Consequently $$[\mathbb{RP}^2,BO(2)] \simeq [\mathbb{RP}^2,\mathbb{CP}^\infty \sqcup \mathbb{CP}^\infty] \simeq [\mathbb{RP}^2,\mathbb{CP}^\infty]\times[\mathbb{RP}^2,\mathbb{CP}^\infty]$$ Using that $\mathbb{CP}^\infty$ is a $K(\mathbb{Z},2)$ we get that $[\mathbb{RP}^2,\mathbb{CP}^\infty]\simeq H^2(\mathbb{RP}^2,\mathbb{Z})\simeq \mathbb{Z}/2\mathbb{Z}$. This would imply that there are $4$ isomorphism classes of circle bundles over $\mathbb{RP}^2$.
Question 1: which characteristic classes give this classification?
Question 2: what are these 4 manifolds $E$?
ATTEMPT 2. This is inspired by Classification of $O(2)$-bundles in terms of characteristic classes. The first Steifel-Whitney class $w_1 \in H^1(\mathbb{RP}^2,\mathbb{Z}/2\mathbb{Z})$ is $0$ iff the bundle is orientable. In this case we have a reduction of the structure group to $SO(2)$ and $SO(2)$ principal bundles are classified by the Euler class $e\in H^2(\mathbb{RP}^2,\mathbb{Z})\simeq [\mathbb{RP}^2, \mathbb{CP}^\infty]\simeq \mathbb{Z}/2\mathbb{Z}.$ Therefore we have $2$ orientable bundles and an unknown number of non-orientable ones when $w_1 = 1\in \mathbb{Z}/2\mathbb{Z} \simeq H^1(\mathbb{RP}^2,\mathbb{Z}/2\mathbb{Z})$.
Question 3: how to find out how many non-orientable bundles we have?
Please tell me if I am correct and if you know, the answer to the above questions.