Let $\pi:X\rightarrow Y$ be a double cover of curves (smooth and projective), and $E$ a vector bundle on $X$ with a non-degenerate symmetric bilinear form $\phi:E\otimes E\rightarrow \mathcal O_X$. With this given, I can associate to it a non-degenerate symmetric bilinear form $$\pi_*\phi:\pi_*E\otimes\pi_*E\rightarrow \pi_*\mathcal O_X$$
My question is , how could one construct $\phi$ starting from $\pi_*\phi$? I mean if given $\psi:\pi_*E\otimes\pi_*E\rightarrow \pi_*\mathcal O_X$, could one construct a $\phi$ s.t $$\pi_*\phi=\psi$$