**Theorem.** Let $X \to Y$ be an étale Galois cover with group $G$ of proper geometrically integral schemes over any field $k$. Then we have an exact sequence
$$0 \to \operatorname{Hom}(G,k^\times) \to \operatorname{Pic}(Y) \to \operatorname{Pic}(X)^G.$$
In particular, the kernel has size at most $n = |G^{\operatorname{ab}}|$, with equality if and only if $n$ is invertible in $k$ and $\mu_r \subseteq k$, where $r$ is the exponent of $G^{\operatorname{ab}}$.

*Proof.* An étale Galois cover with group $G$ is the same thing as a $G$-torsor, and we have a Hochschild–Serre spectral sequence (see e.g. Milne's *Étale cohomology* notes, **Thm 14.9**)
$$E_2^{pq} = H^p(G,H^q(X,\mathbb G_m)) \Rightarrow H^{p+q}(Y,\mathbb G_m).$$
The exact sequence of low degree terms is
$$0 \to H^1(G,\mathbb G_m(X)) \to H^1(Y,\mathbb G_m) \to H^1(X,\mathbb G_m)^G \to H^2(G,\mathbb G_m(X)) \to \ldots .$$
Since $X$ is proper and geometrically integral, the global sections of $\mathcal O_X$ are just the constants $k$, hence the global sections of $\mathbb G_m$ are just $k^\times$. This clearly has the trivial $G$-action, so
$$H^1(G,\mathbb G_m(X)) = \operatorname{Hom}(G,k^\times).$$
This proves the first statement. The second statement follows since
$$\operatorname{Hom}(G,k^\times) = \operatorname{Hom}(G^{\operatorname{ab}},k^\times),$$
and for any finite abelian group $G$ there is a noncanonical isomorphism $G^{(p')} \cong \operatorname{Hom}(G,\bar k^\times)$, where $(-)^{(p')}$ denotes the prime to $p$-part if $\operatorname{char} k = p > 0$ (and the entire group if $\operatorname{char} k = 0$).

Since the group generated by the images of homomorphisms $G \to \bar k^\times$ is the subgroup $\mu_r$ for $r$ the exponent of $G$, we conclude that they are all realised over $k$ if and only if $\mu_r \subseteq k$. $\square$