Let us consider $R_n = \mathbb Z_\ell[\zeta_{\ell^n}]$, an auxiliary prime power $q\equiv 1 \pmod \ell$ and $R_0$ to be the fixed field of the automorphism $\zeta_{\ell^n} \to \zeta_{\ell^n}^q$. (If I am not mistaken, this is independent of $n$ at least for large $n$).

Is there a nice description of the Galois cohomology groups $H^1(\operatorname{Gal}(R_n/R_0), GL_k(R_n))$? At least for $k = 1$? Surely this is very standard but I am having a hard time finding references.

Let us consider $R_n = \mathbb Z_\ell[\theta_n]/(\theta^{\ell^n}-1)$, an auxiliary prime power $q\equiv 1 \pmod \ell$ with an action of $\mathbb Z = \langle \sigma\rangle$ by $\sigma(\theta_n) = \theta_n^q$. It acts through a quotient $\mathbb Z/\ell^{n-n_0}$ for some $n_0$.

Is there a nice description of the cohomology groups $H^1(\mathbb Z, GL_k(R_n))$? At least for $k = 1$? I am having a hard time finding references.

(The question has been edited after I realized that I wanted to solve a slightly different problem.)