Decomposability of extremal positive unital maps on $M_2$

A map $\varphi :M_m(\text{bC})\to M_n(\text{bC})$ is decomposable if it is of the form $\varphi ={\varphi }_1+{\varphi }_2$ where ${\varphi }_1$ is a CP map while ${\varphi }_2$ is a co-CP map. It is known that if $m=n=2$ then every positive map is decomposable. Given an extremal unital positive map $\varphi :M_2(\text{bC})\to M_2(\text{bC})$ we construct concrete maps (not necessarily unital) ${\varphi }_1$ and ${\varphi }_2$ which give a decomposition of $\varphi$. We also show that in most cases this decomposition is unique.