Let $X$ be a scheme over $k$ and $p:\ X \to Spec(k)$ the structure morphism. If $M$ is an étale sheaf of abelian groups over $Spec(k)$ I have a Grothendieck spectral sequence $$E^{p,q}_2=Ext^p_k(M,R^qp_*\mathbb{G}_m)\Rightarrow Ext^{p+q}_{X_{et}}(p^*M,\mathbb{G}_m),$$ obtained composing the functors $p_*$ and $Hom_k(M,-)$ (the last functor goes to $Ab$). Now if $0\to A \to B \to C \to 0$ is an exact sequence of étale sheaves of abelian groups over $X$, such that the corresponding sequence $$0\to p_*(A) \to p_*(B) \to p_*(C) \to R^1p_*(A)\to 0$$ is exact, then the transgression map (the map $E_2^{0,1} \to E_2^{2,0} $) of the spectral sequence $$Hom_k(-,R^1p_*(A))\to Ext^2_k(-,p_*(A))$$ should be the inverse of the Yoneda pairing with $$0\to p_*(A) \to p_*(B) \to p_*(C) \to R^1p_*(A)\to 0.$$
I'm looking for a proof of this fact.
