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.