Let $k$ be a perfect field of characteristic $p>0$, $W=W(k)$ its ring of Witt vectors, $K_0=W(k)[\frac{1}{p}]$ and, $K/K_0$ be a totally ramified extension. Let $\pi \in K$ be an uniformizer.

Consider the Kummer/Breuil extension $K_{\infty}:=\bigcup_{n \geq 0} K_n$, where $K_n:=K(\pi_n)$ and define $G_{K_{\infty}}=\text{Gal}(\bar K/K_{\infty}) \subset G_K=\text{Gal}(\bar K/K)$, where $\bar K$ is the algebraic closure of $K$.

**My question:**

Denote by $\text{Rep}_{\mathbb{Q}_p}(G_{K_\infty})$ the category of continuous representations of $G_{K_\infty}$ on finite dimensional $\mathbb{Q}_p$-vector spaces or in other words, $\text{Rep}_{\mathbb{Q}_p}(G_{K_\infty})$ is the category of $p$-adic representations of $G_{K_\infty}$. Also denote by $\text{Rep}_{\mathbb{Z}_p}(G_{K_\infty})$ is the category of $\mathbb{Z}_p$-representations. Now $p$-adic Tate module $T_p(G)$ of $p$-divisible group $G$ is equipped with structure of $\mathbb{Z}_p$-module. \begin{align} &\text{(1) When $T_p(G) \in \text{Rep}_{\mathbb{Q}_p}(G_{K_\infty}) \ \text{or} \ \text{Rep}_{\mathbb{Z}_p}(G_{K_\infty})$ ?} \\ &\text{(2) What is the relation between the category $\text{Rep}_{\mathbb{Q}_p}(G_{K_\infty})$ and the category } \\ & \hspace{2cm} \text{ of $p$-adic Tate modules $T_p(G)$ of $p$-divisible groups $G$ ?} \\ & \text{(3) Is the functor $\text{Rep}_{\mathbb{Z}_p \ or \ \mathbb{Q}_p}(G_{K_{\infty}})$} \longrightarrow \{\text{category of} \ T_p(G) \} \ full \ or \ surjective ? \end{align}

I am trying to identify $T_p(G)$ in the category $\text{Rep}_{\mathbb{Q}_p}(G_{K_\infty})$ or in $\text{Rep}_{\mathbb{Z}_p}(G_{K_\infty})$.

**Case I:** When we consider $G$ is just the abelian group $\text{of roots of unity}$ in the separable closure $K^s$ of $K$, then the $p$-adic Tate module is the rank one free $\mathbb{Z}_p$-module equipped with action of the absolute Galois group $G_K$ of $K$. That is, in this case the $p$-adic Tate module is the Galois representation or $p$-adic cyclotomic character of $K$.

**Case II:** When $G$ is the abelian variety over the field $K$, the $K^s$-valued points of $G$ form an abelian group. The $p$-adic Tate module $T_p(G)$ of $G$ is the Galois representation of the absolute Galois group $G_K$ of $K$.

But,more generally, I am trying to identify $T_p(G)$ in the category $\text{Rep}_{\mathbb{Q}_p}(G_{K_\infty})$ or in $\text{Rep}_{\mathbb{Z}_p}(G_{K_\infty})$.

Any help would be appreciated.