# sequences in non-abelian group cohomology

In general, if we have a (pro-)finite group $$G$$ and a sequence of (continuous) non-abelian $$G$$-modules $$1\rightarrow A\rightarrow B\rightarrow C\rightarrow 0,$$ such that the image of $$A$$ lies in the center of $$B$$ we get a "long" exact sequence of pointed sets $$\dots\rightarrow H^1(G,A)\rightarrow H^1(G,B)\rightarrow H^1(G,C)\overset{\Delta}{\rightarrow}H^2(G,A).$$ It seems to me that in general you cannot say something about the image of $$\Delta$$, but I want to ask, if you can say something about the image of $$\Delta$$ in the following setup:

Let $$k$$ be a finite field with $$\#k=q=p^f$$ and denote by $$k_E=k((X))$$ its field of Laurent series. This is a local field of characteristic $$p$$ with residue field $$k$$. Fix a seperable algebraic closure of $$k_E$$ denoted by $$k_E^{sep}$$. Let $$G=G_{k_E}=Gal(k_E^{sep}|k_E)$$ be the absolute Galois group of $$k_E$$. We view $$k\subset k_E$$ and so as a trivial $$G$$-module.

By local class field theory we have that $$H^2(k_E)=H^2(G,(k_E^{sep})^\times)\cong\mathbb{Q}/\mathbb{Z}$$ and via the short exact sequence $$1\rightarrow k^\times\overset{\subset}{\rightarrow}(k_E^{sep})^\times \overset{(\cdot)^{q-1}}{\rightarrow}(k_E^{sep})^\times\rightarrow 1$$ we can compute $$H^2(G,k^\times)\cong\frac{1}{q-1}\mathbb{Z}/\mathbb{Z}\subset\mathbb{Q}/\mathbb{Z}$$. Now consider the short exact sequence of trivial $$G$$-modules $$1\rightarrow k^\times\rightarrow GL_n(k)\rightarrow PGL_n(k)\rightarrow 1.$$ I want to find out what the image of $$\Delta:H^1(G,PGL_n(k))\rightarrow H^2(G,k^\times)\cong\mathbb{Z}/(q-1)\mathbb{Z}$$ is. On the other hand consider the following sequnce of $$G$$-modules given by the natural action: $$1\rightarrow (k_E^{sep})^\times\rightarrow GL_n(k_E^{sep})\rightarrow PGL_n(k_E^{sep})\rightarrow 1.$$ It is known that this gives us $$H^1(G,PGL_n(k_E^{sep}))\cong\frac{1}{n}\mathbb{Z}/\mathbb{Z}\subset\mathbb{Q}/\mathbb{Z}$$ via the linking morphism $$\Delta$$ [See Serre, Local fields chapter X, §5 Lemma 1]. So functoriality of the linking morphism gives us that $$\Delta:H^1(G,PGL_n(k))\rightarrow\mathbb{Z}/(q-1)\mathbb{Z}$$ is constant, if $$n$$ and $$q-1$$ are coprime. But, on the other extreme, what if we have $$n=q-1$$ (eg. n=2, q=3)? Might $$\Delta:H^1(G,PGL_n(k))\rightarrow \mathbb{Z}/(q-1)\mathbb{Z}$$ then even be surjective?