Let $X=\mathbb{T} \times \mathbb{RP}^2$. I want to find all covering scpaces (up to isomorphism) of $X$.
First, suppose we know all covering spaces of toro $\mathbb{T}$ and projective plane $\mathbb{RP}^2$. We know that if $(\tilde A,p)$ is a covering space of $\mathbb{T}$ and $(\tilde B,q)$ is a covering space of $\mathbb{RP}^2$, then $(\tilde A \times \tilde B, p \times q)$ is a covering space of $X$. By this information we can find a universal covering space of $X$ (for example $\mathbb{R}^2 \times \mathbb{S}^2$) and so we can conclude that all covering spaces of $X$ are a quotient between the universal covering space and subgroups of $\pi_1(X)=\mathbb{Z}^2 \times (\mathbb{Z}/2\mathbb{Z})$ which is abelian. Using this corrispondence we observe again that all covering sapces in the form $(\tilde A \times \tilde B, p \times q)$ are associated with subgroups $A \times B < \mathbb{Z}^2 \times (\mathbb{Z}/2\mathbb{Z})$, but this is not enough to complete the list because there exist subgroups of $\mathbb{Z}^2 \times (\mathbb{Z}/2\mathbb{Z})$ which are not product of a subgroup of $\mathbb{Z}^2$ with once of $\mathbb{Z}/2\mathbb{Z}$. How can we actually find out the covering spaces of $X$ linked to these particular subgroups?
(Stupid question: Is true that all covering spaces of $\mathbb{T}$ are also of $X$? I think so, but it looks quite weird...)