Let $f:X\to Y$ be a finite surjective morphism of smooth integral projective varieties over an algebraically closed field $k$ of characteristic 0. Denote by $CH_i(W):=Z_i(W)/\sim$ the Chow group of $i$-cycles on $W$, where $Z_i(W)$ is the group of $i$-cycles on $W$ and $\sim$ means rational equivalence. The proper push-forward $f_\ast:Z_i(X)\to Z_i(Y)$ is a group homomorphism that is compatible with rational equivalence, meaning that it descends to a group homomorphism $f_\ast:CH_i(X)\to CH_i(Y)$. When $i=n−1$, under the above assumptions on $X$ and $Y$ we have that $CH_{n−1}(X)=Pic(X)$ and $CH_{n−1}(Y)=Pic(Y)$. Hence, $f_\ast$ is a group homomorphism $f_\ast:Pic(X)\to Pic(Y)$. Assuming that $Pic(Y)$ and $Pic(X)$ are representable by group schemes (which they are, under our assumptions), my question is:

Is $f_\ast:Pic(X) \to Pic(Y)$ a morphism of group schemes?