Complementation in the Group of Units of Matrix Rings

Let $R$ be a ring with $1$ and $\text{J}(R)$ its Jacobson radical. Then $1+\text{J}(R)$ is a normal subgroup of the group of units, $G(R)$. The existence of a complement to this subgroup was explored in a paper by Coleman and Easdown; in particular the ring $R={\text{Mat}}_n({\mathbb{Z}}_{p^k})$ was considered. We prove the remaining cases to determine for which $n$, $p$ and $k$ a complement exists in this ring.