Let $X=Jac(C)$ be an abelian surface over $\mathbb{C}$, the Jacobian of a genus 2 curve. Let $L$ be a symmetric line bundle. Let $Y$ be the Kummer surface, quotient of $X$ by the action of involution. Then $L^2$ is totally symmetric, hence there is a line bundle $L'$ on $Y$ which pulls back to $L^2$. Further since $L^2$ embeds $Y$ in $\mathbb{P}^3$, $L'$ is very ample.

1) Is $L'$ the unique line bundle which pulls back to $L^2$?

2) Consider a curve $C\in |L^2|$, which is smooth, preserved under involution and avoiding the 16 double points of $X$. Then the image of $C'$ is smooth and avoids the 16 singular points of $Y$. Can we say that $C'\in |L'|$?

Thanks!