Let $k$ be a finite field and $k_E:=k((X))$ denote the field of Laurent series over $k$. We define a Frobenius endomorphism on $k_E$ via $f(X)\mapsto f(X^p)$. We choose a lift $\varphi:k_E^{sep}\rightarrow k_E^{sep}$ on the seperable closure of $k_E$ (eg. if $k=\mathbb{F}^p$, we can choose $\varphi:=(\cdot)^p$).

Now let $\mathbb{G}$ be a linear algebraic group over $k$ and let $\mathbb{N}$ act on $\mathbb{G}(k_E^{sep})$ via $1_{\mathbb{N}}\mapsto \mathbb{G}(\varphi)$. I want to know, if there are nice classes of such $\mathbb{G}$ (eg. reductive $\mathbb{G}$ would be great :P), such that $$H^1(\mathbb{N},\mathbb{G}(k_E^{sep}))=1,$$ i.e. for every $A\in\mathbb{G}(k_E^{sep})$ there exists a $B\in\mathbb{G}(k_E^{sep})$, such that $B=A\cdot\mathbb{G}(\varphi)(B)$.

For $\mathbb{G}=GL_n$ this seems to be true, since for a finite dimensional $k_E^{sep}$-vector space $V$ equiped with a $\varphi$-semilinear map $\varphi_V$, there always exists a $k_E^{sep}$-basis $(v_i)_i$, such that $\varphi_V(v_i)=v_i$. See [Peter Schneider: Galois representations and $(\varphi,\Gamma)$-modules, Proposition 3.2.4] or if you are capable to read some german [https://ivv5hpp.uni-muenster.de/u/pschnei/publ/lectnotes/Theorie-des-Anstiegs.pdf, Satz 2.1].