From $0\rightarrow \mathbb{Z}\rightarrow \mathbb{Z}\rightarrow \mathbb{Z}_{2}\rightarrow 0$, we have the long exact sequence $H^{1}(X,\mathbb{Z})\rightarrow H^{1}(X,\mathbb{Z}_{2})\rightarrow H^{2}(X,\mathbb{Z})$. Meanwhile $H^{1}(X,\mathbb{Z}_{2})\cong H^{1}(X,\mathbb{Z})\otimes\mathbb{Z}_{2} \oplus Tor(H^{2}(X,\mathbb{Z}),\mathbb{Z}_{2})$.

My question is: if $L$ is a real line bundle on $X$, does $w_{1}(L)$ have component in $H^{1}(X,\mathbb{Z})\otimes\mathbb{Z}_{2}$ or $Tor(H^{2}(X,\mathbb{Z}),\mathbb{Z}_{2})$ under the above isomorphism.