Automorphisms of the Gersten group

The Gersten group $G$ is the split extension $F_3\rtimes_\varphi{\Bbb Z}$ of the free group $F_3$ with basis $\{a,b,c\}$ by the automorphism $\varphi: a\mapsto a, b\mapsto ba, c\mapsto ca^2$. We describe the generators and structure of the group $\operatorname{Out}(G)$ and prove that $\operatorname{Out}(G)\cong(F_3\times{\Bbb Z}^3)\rtimes({\Bbb Z}_2\times{\Bbb Z}_2)$.