such types has an analogous argument.

Since each $\lambda_i$ can be written as $e^{i\theta_i}$, for some $\theta_i \in {\mathbb{R}}$, we can then see that $(\lambda_0,\ldots, \lambda_n) \in {\text{Aut}}(F_A)$ if and only if we have that $\prod_{j=0}^n e^{ia_{ij}\theta_j} = 1$ for all $i$. The map $(\lambda_0, \ldots, \lambda_n) \mapsto (\frac{1}{2\pi i} \log (\lambda_0), \ldots, \frac{1}{2\pi i} \log(\lambda_n))$ induces an isomorphism $$\label{autisom}
{\text{Aut}}(F_A) \cong \left\{ (z_0,\ldots, z_n) \in ({\mathbb{R}}/{\mathbb{Z}})^n\text{ } \middle| \text{ }A \left(\begin{array}{c}z_0 \\ \vdots \\ z_n\end{array}\right) \in {\mathbb{Z}}^{n+1} \right\}.$$ We then observe that we can describe ${\text{Aut}}(F_A)$ as being generated by the elements $$\rho_i = (e^{2\pi i b_{0i}/d}, \ldots, e^{2\pi b_{ni}/d}) \in ({\mathbb{C}}^*)^{n+1}.$$ Moreover, there is a characterization by Artebani, Boissière, and Sarti of the group ${\text{Aut}}(F_A)$ (Proposition 2 of [@ABS]):

$\emph{Aut}(F_A)$ is a finite abelian group of order $|\det A|$. If we think of $F_A$ as a sum of atomic types, $F_{A_1}(x_0, \ldots, x_{i_1}) + \ldots + F_{A_k}(x_{i_{k-1}+1}, \ldots, x_n)$, then we may characterize the elements of $\emph{Aut}(F_A)$ as being the product of the $k$ groups $\emph{Aut}(F_{A_i})$. The groups $\emph{Aut}(F_{A_i})$ are determined based on the atomic types:

1.  For a summand of Fermat type $W_{\text{Fermat}} = x^a$, the group $\emph{Aut}(W_{\text{Fermat}})$ is isomorphic to ${\mathbb{Z}}/a{\mathbb{Z}}$ and generated by $\varphi = 