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?

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