A way to realize the $H^1(B;\mathbb{Z}/2)$ action on Pin- and Spin- structure

Question

A Pin-structure on a real vector bundle $F^n\to B$ is a $Pin_n$ principal bundle $P^\#\to B$ with an isomorphism $P^\#\times_{Pin_n}\mathbb{R}^n\cong F$. A familiar property shows that $H^1(B;\mathbb{Z}/2)$ acts freely and transitively on Pin-structures and Spin-structures. In Paul Seidel's book Fukaya Categories and Picard-Lefschetz Theory, he says that given a Pin-structure and a real line bundle $\beta\to B$ the fibre product $P^\#\times_{\mathbb{Z}_2}S(\beta)$ can form a new Pin-structure where $\mathbb{Z}/2$ acts by $\pm1$ on the double covering $S(\beta)$ associated to $\beta$ and by $\{\pm e\}$ on $P^\#$. He denotes the new structure by $P^\#\otimes \beta$. Since line bundles are classified by $H^1(\mathbb{Z}/2)$, this construction realizes the action of $H^1$. However, I cannot fully understand this fibre product, to be more specific, its transition data and its action over $F$ or the frame bundle. Furthermore, I want to apply this kind of construction to Spin-structures. The construction is the same, but I don't know how to relate the fibre product to the orientation of the original Spin-structure. Do you have any ideas on how to view this fibre product? In particular, is there a geometric way to see it if $B$ is a loop and $n=1$? Thanks.