Problem related to semidirect product

Question

I have a small question regarding the semidirect product. Consider a group $G$ which is the semidirect product $\mathbb{Z}_3 \ltimes (\mathbb{Z}_5 \times \mathbb{Z}_5)$ (internal semidirect product). Let $\phi\colon\mathbb{Z}_3\to\operatorname{Aut}(\mathbb{Z}_5 \times \mathbb{Z}_5)$. There will be $\phi_0 ,\phi_1 ,\phi_2$ corresponding to $\bar{0},\bar{1},\bar{2}$ of $\mathbb{Z}_3$, right?

Answer 1

Since Aut$(\mathbb{Z}_5)=C_4$, Aut($\mathbb{Z}_5 \times \mathbb{Z}_5$) $\cong C_4\times C_4$. The only action $\mathbb{Z}_3$ can have on $\mathbb{Z}_5 \times \mathbb{Z}_5$ is the trivial one. Thus you get a direct product in the end.