Let $G$ be a $p$-divisible group over the ring of $p$-adic integers $O_K$ of $p$-adic field $K$.
The $p$-adic Tate module $T_p(G)$ of $G$ is rank $1$ free $\mathbb{Z}_p$-module.
Then $T_p(G) \otimes_{\mathbb{Z}_p} \mathbb{Q}_p$ is a vector space over $\mathbb{Q}_p$.
Let $A$ is a finite $\mathbb{Z}_p$-algebra, then consider the tensor product $T_p(G) \otimes_{\mathbb{Z}_p}A$.
My question:
What about the algebraic structure of $T_p(G) \otimes_{\mathbb{Z}_p}A$ ?
$(1)$ Is it a free $A$-module ?
$(2)$ Can we think of $T_p(G) \otimes_{\mathbb{Z}_p}A$ as a Galois representation of some suitable Galois group?
Any help with two questions.
Edit: For $(2)$, one exmaple is as follows: If $G$ is the abelian group of roots of unity in a separable closure $K^s$ of $K$,then the $p$-adic Tate module $T_p(G)$ is equipped with linear action of the absolute Galois group of $K$ or $T_p(G)$ is the Galois representation or more specifically, $p$-adic cyclotomic character.
But what about if $G$ is the$p$-divisible group ?