Uniqueness and Universality of the Ring $W_n(K_s)$ of Truncated Witt Vectors in generalized Kummer Theory

It is well known that for a finite field $K$ of characteristic $p$ the ring of $n$-truncated Witt vectors $W_n(K)$ is used to classify field extensions of $K$ of degree $p^n$; for details see e.g. Bosch's Algebra, chapter 4.10 on general Kummer theory.

Basically the upshot is, cyclic subgroups $\Delta \subseteq W_n(K) / \wp(W_n(K_s))$ of order $p^n$ correspond to extensions $K(\wp^{-1}(\Delta))/K$, where $\wp: x \mapsto F(x)-x$ (note the algebraic operations happen in $W_n(K_s)$) is a surjective operator on $W_n(K_s)$.

Now Bosch's Theorem 1 on p. 208 suggests (at least how I understand this part) that seemingly the ring of truncated Witt vectors is not so special in its role to classify such $p^n$-extensions of $K$. Essentially every $G:=\text{Gal}(K)$-module $A$ together with a surjective $G$-homomorphism $\wp: A \to A$, whose kernel $\mu_n$ in $A_K := A^{\text{Gal}(K_s/K)}$ is a finite cyclic subgroup in $A_K$ of order $p^n$ and which satisfies for every cyclic Galois extension $L/K$ of degree dividing $p^n$ the cohomological condition $H^1(\text{Gal}(L/K), A_L) = 0$, classifies such extensions of exponent $p^n$.

Therefore the Question: Is the ring $A:=W_n(K_s)$ of $n$-truncated Witt vectors regarded as $G$-module unique (to which extent) or distinguished in an appropriate sense with these properties, i.e. with:

• $H^1(\text{Gal}(L/K), A_L) = 0$ for all cyclic Galois extension $L/K$ of degree dividing $p^n$
  -admitting surjective $G$-homomorphism $\wp: A \to A$, whose kernel $\mu_n$ in $A_K := A^{\text{Gal}(K_s/K)}$ is a finite cyclic subgroup in $A_K$ of order $p^n$

If not, is $W_n(K_s)$ at least universal in appropriate sense under $G$-modules with these two properties?

If yes, are there references?

Follow up question: Moreover, the non-truncated ring of Witt vectors posesses (see e.g. Dongryul Kim's blog post) another number-theoretic universal property: If we take a finite field $K:=\Bbb F_q$ with $q=p^n$ elements, it outputs a ring $W(K)=O_L$, where $L$ is the unique unramified extension of $\Bbb Q_p$ of degree $n$.

Is this universal property somehow directly related to its role in classification of generalized Kummer theory?