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.)