Let $G$ be an affine groups scheme over $\mathbb Z$. As such it has an associated Hopf algebra, $A=\mathbb Z[G]$ such that $G(R)$ is naturally identified with the set $\hom_{Rng}(A,R)$ of ring homomorphisms, where the group operations (multiplication, inverse, unit) are given on this set from the co-operations of the algebra $A$. 

Fix $m\in\mathbb{N}$ and $p$ a prime number, and let $W_m$ be the functor of $p$-typical Witt vectors of length $m+1$. The functor $W_m$ is represenatble as well, with representing algebra $\mathbb{Z}[x_0,\ldots,x_m]$.

I am interested in the structure of the group scheme $R\mapsto G(W_m(R))$. Greenberg's results imply that this functor is a an affine group scheme as well, and hence representable. Is there any known construction for the associated Hopf-algebra of $G\circ W_m$?

____
Some obvious facts-

In the case where the prime $p$ is invertible in $R$, the Witt ring $W_m(R)$ is isomorphic to the product ring $\prod_{i=0}^m R$. Since the group scheme $R\mapsto G(\prod_{i=0}^{m} R)=\prod_{i=0}^{m}G( R)$ is represented by the Hopf algebra $A^{\otimes m+1}=\underbrace{A\otimes\cdots\otimes A}_{m+1\text{-fold}}$, I somehow expect there to exist a map between the representing algebra of $G\circ W_m$ and $A^{\otimes m+1}$, which becomes an isomorphism under localization by $p$. 

On the other hand, in the complementary case, the group $G(W_m(R))$ is usually nothing like $G(R)\times G(R)$. For example, in the case $G= GL_n$, $R=\mathbb{F}_p$ and $m=2$, we have an exact sequence
$$1\to M_n(R)\to GL_n(W_2(R))\to GL_n(R)\to 1.$$
In particular, $|GL_n(W_2(R))|\ne |GL_n(R)\times GL_n(R)|$, and one cannot expect any sort of bijection to exist between the two.

____
I'm new to the subject, and my foundations on algebraic groups or algebraic geometry are not incredibly solid, so I apologize if anything I wrote above does not make complete sense, or is obviously false. 

I would very much appreciate any clue or reference to the construction of the Hopf algebra of $G\circ W_m$, or any other interesting facts regarding the structure of G\circ W_m$. 

Thank you very much!
Shai