Let $X$ be a scheme (you can assume that $X$ is proper and smooth over an algebraically closed field) and $T$ is a finite subgroup of $\text{Pic } X$ (of order prime to the characteristic). Does there exist a finite étale cover $f:Y\to X$ (possibly Galois with Galois group $T$) for which the map $f^\* :\text{Pic }X\to\text{Pic }Y$ identifies $\text{Pic } Y$ with the quotient $(\text{Pic }X)/T$?


Of course (except for the claim about $f$ being Galois) it suffices to treat the case when $T$ is cyclic (of order $m$ say). Then we take a generator $L$ of $T$ and 
$$ Y = \text{Spec} {}\_X \bigoplus\_{i=0}^{m-1} L^i $$
(the standard construction showing that $H^1(X, \mu\_m)$ classifies such covers). I
think I can show that the kernel of $\text{Pic }X\to\text{Pic }Y$ is $T$, but I don't
know how to prove that it is surjective...


It also reminds me of the Hilbert class field: I don't know much
algebraic number theory, but I think that the construction is similar:
$K$ is a number field, $X = \text{Spec }\mathcal{O}\_K$, $T = \text{Pic }X$ is the class group, then $Y = \text{Spec }\mathcal{O}\_H$ where $H$ is the Hilbert class field of $K$, its maximal abelian unramified extension.



Picard groups of abelian étale covers
The answer is "no," even for curves. Say $X$ is a genus 2 curve over $\mathbf{C}$, and $f:Y \to X$ is any finite etale degree $d$ morphism of degree $> 1$. Then $Y$ is a smooth projective curve of genus $d+1 > 2$. In particular, $\mathrm{Pic}(Y)$ has dimension $>2$, so it cannot be a quotient of $\mathrm{Pic}(X)$.


