Let $x$ be the closed point of an $n$-dimensional local scheme $X$, essentially smooth over a field $k$. Let $M$ be a sheaf on $X$ (in either Zariski or Nisnevich topology). I came across the following implicit claim in what I am currently reading :

Claim: Automorphisms of $X$ which acts trivially on the residue field $k(x)$ of $x$, act trivially on $H^n_x(X,M)$,

where $H^n_x(X,M)$:= cohomology with supports along $x$.

I think the above claim should be false. I am looking for a counter example (or proof!).

Let $x$ be the closed point of an $n$-dimensional local scheme $X$, essentially smooth over a field $k$. Let $M$ be a sheaf on the category of smooth $k$-varieties (in either Zariski or Nisnevich topology). I came across the following implicit claim in what I am currently reading :

Claim: Automorphisms of $X$ which acts trivially on the residue field $k(x)$ of $x$, act trivially on $H^n_x(X,M)$,

where $H^n_x(X,M)$:= cohomology with supports along $x$.

I think the above claim should be false. I am looking for a counter example (or proof!).