Disclaimer: I'm not an expert in étale cohomology. This just came up, when trying to generalize a method, that came up studying a problem in algebraic number theoy/ class field theoy.
Let $f:Y\to X$ be a Galois cover of schemes. Then for an étale sheaf $\mathcal{F}$ on $X$ we have the Hochschild-Serre spectral sequence $$ E^{p,q}_2=H^p(G,H^q_{et}(Y,\mathcal{F}|_Y))\Rightarrow H_{et}^{p+q}(X,\mathcal{F}). $$ From the usual five term sequence associated to a spectral sequence, we get the exact sequence $$ 0\to H^1(G,\mathcal{F}(Y))\overset{\varphi}\to H^1_{et}(X,\mathcal{F})\overset{\psi}\to H^1_{et}(Y,\mathcal{F}|_Y)^G. $$ With the interpretation $H^1(G,\mathcal{F}(Y))=\check{H}^1(\{Y\to X\},\mathcal{F})$ it is not hard to explicitly describe the first map. The second map is also not hard to describe in terms of Cech cohomology, but I'm struggling to see show the exactness at $H^1_{et}(X,\mathcal{F})$ in terms of these maps.
Question: Given a class $\alpha\in H^1_{et}(X,\mathcal{F})$, that vanishes in $H^1_{et}(Y,\mathcal{F}|_Y)$, how can I explicitly construct a $1$-cocycle $\gamma\in H^1(G,\mathcal{F}(Y))$ such that $\varphi(\gamma)=\alpha$?
I'm probably missing something obvious, but I've been trying for some days now.
Any help is greatly appreciated!
The background to my question: I'm particularly interested in the case, where $\mathcal{F}=\mathbb{G}_m$. I have a map defined from a subgroup $A$ of $\mathrm{Pic}(Y)$ to $ \mathrm{Pic}(X)$, whose image is in the kernel of $\mathrm{Pic}(X)\to \mathrm{Pic}(Y)$ and need that the induced map $A\to H^1(G,\mathbb{G}_m(Y))$ sits in a certain commutative diagram. I don't see another way to show the commutativity except to describe this map explicitly.
If you want I can provide more details to the motviation and concrete setting.
