Let $p:S^1\times S^3\rightarrow S^1\times S^3$ be a covering map with $p(z,y)=(z^3,y)$ and $z\in S^1\subset\mathbb{C}$ and $h:\mathbb{R}P^4\rightarrow S^1\times S^3$. Is there a lift $g:\mathbb{R}P^4\rightarrow S^1\times S^3$ with $pg=h$?
I just noticed that I have to show if $g_*(\pi_1(\mathbb{R}P^4))=g_*(\mathbb{Z}/2)\subset p_ * (\mathbb{Z})=p_*(\pi_1(S^1))$ is true or false.