Let $X$ be a smooth, projective variety, $Y \subset X$ a smooth, projective subvariety of codimension $3$. Denote by $\pi:\tilde{X} \to X$ the blow-up of $X$ along $Y$ and by $E$ the exceptional divisor. We know, that $\pi|_E:E \to Y$ is a $\mathbb{P}^2$-bundle. Let $\pi':F \to Y$ be a $\mathbb{P}^1$-sub-bundle of $E$ in the sense that there exists a closed immersion $i:F \hookrightarrow E$ such that $\pi|_E \circ i=\pi'$. Is it possible to say something about the kernel of the push-forward morphism $i_*:\mathrm{Pic}(F) \to A^2(E)$ i.e., about its generators, where $A^2(E)$ denotes the Chow group of codiension $2$ cycles on $E$. If necessary assume that $\mathrm{Pic}(Y)=\mathbb{Z}$.

I know that this is not the Gysin map on Chow groups. The question is motivated by Gysin map on cohomology, rather its restriction to certain Hodge structures. To simplify the original question, I have given the above formulation.