The construction I am interested in is the following:

Let $V$ be a complete discrete valuation ring of mixed characteristic $(0,p)$ with algebraically closed residue field $k$. Let $K=Frac(V)$ and fix an algebraic closure $\overline{K}$ of $K$, and $\overline{V}$ the integral closure of $V$ in $\overline{K}$. Define $R_{\overline{V}}$ to be the perfection $\lim_{\leftarrow \, Frob} \overline{V}/p\overline{V}$; take the Witt ring $W(R_{\overline{V}})$ one gets a surjective homomorphism $$\theta: W(R_{\overline{V}}) \rightarrow \widehat{\overline{V}}$$ by $\theta([x])= \lim_{\rightarrow \infty} \widetilde{x_m}^{p^m}$ where $x=(x_n)_{n\in \mathbb{N}} \in R_{\overline{V}}$ $\widetilde{x_m}$ a Lifting of $\overline{V}/p\overline{V}$ to $\overline{V}$ and $[\cdot]$ the Teichmüller representative. As is well known the kernel of $\theta$ is generated by a single element $\xi$. From here I am interested in understanding $A_N(\overline{V}):= W(R_{\overline{V}})/\xi^N W(R_{\overline{V}})$.

Now to my question: I want to see what comes out of this construction when I put in $W(k)=V$. At first I thought that it was quite simple because the projection on the first factor $R_{W(k)} \overset{\sim}{\rightarrow} k$ is an isomorphism which yields the isomoprhism $W(R_{W(k)}) \overset{\sim}{\rightarrow} W(k)$ but this does not account for the integral closure being taken in $\overline{K}=\overline{Frac(W(k))}$. So my question boils down to: what is the integral closure $\overline{W(k)}$ of $W(k)$ in $\overline{K}$ and how does above construction work for $\overline{W(k)}$?