The Gersten group \( G \) is the split extension \( F_{3}\rtimes_{\varphi}{𝕑} \) 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 \( \operatorname{Out}(G)\cong(F_{3}\times{𝕑}^{3})\rtimes({𝕑}_{2}\times{𝕑}_{2}) \).

中文翻译：

---

Gersten 群的自同构

Gersten 群\( G \)是自由群\( F_{3} \)的分裂扩展 \( F_{3}\rtimes_{\varphi}{𝕑} \)基\( \{a,b ,c\} \)由自同构\( \varphi:a\mapsto a,b\mapsto ba,c\mapsto ca^{2} \)。我们描述了群的生成器和结构 \( \operatorname{Out}(G) \) 并证明 \( \operatorname{Out}(G)\cong(F_{3}\times{𝕑}^{3} )\rtimes({𝕑}_{2}\times{𝕑}_{2}) \) .