Linear representation of finite group over finite field of coprime characteristic lifts uniquely to Witt ring

For ring of Witt vectors truncated to a finite length
Suppose $$G$$ is a finite group and $$K$$ is a finite field whose characteristic is a prime number $$p$$ not dividing the order of $$G$$. Suppose $$W$$ is the fact about::ring of Witt vectors over $$K$$ of length $$d$$. Then, for any representation $$\varphi:G \to GL(n,K)$$, there exists a representation $$\tilde{\varphi}: G \to GL(n,W)$$ that reduces to $$\varphi$$ under the map $$GL(n,W) \to GL(n,K)$$ induced by the quotient map $$W \to K$$.

Further, $$\tilde{\varphi}$$ is unique up to equivalence of representations.

When $$K$$ is the prime field $$\mathbb{F}_p$$, $$W$$ is the ring $$\mathbb{Z}/p^d\mathbb{Z}$$. When $$K$$ is a finite field $$\mathbb{F}_{p^r}$$, $$W$$ is a Galois ring of length $$d$$ with residue field of order $$p^r$$.

Facts used
This uses the two parts of the uses::Schur-Zassenhaus theorem:


 * 1) uses::Normal Hall implies permutably complemented: This justifies existence.
 * 2) uses::Hall retract implies order-conjugate: This justifies uniqueness up to equivalence.