The paper I am referencing is "Normal Subgroups of the Cremona Group." https://arxiv.org/abs/1007.0895. In theorem 5.14, at the bottom of page 52, the author stated for the abelian surface $Y= \mathbb{C}/\mathbb{Z}[i] \times \mathbb{C}/\mathbb{Z}[i]$, we consider the quotient $X= Y/ \Gamma$ where $\Gamma$ is the group of order 4 generated by the map $(x,y)\mapsto (ix, iy)$ on $Y$. Moreover one can show that $X$ is rational.

In general the group of automorphism of $X$ cannot be lifted to that on $Y$ since there are more automorphisms on a rational surface than on an abelian one. However, the paper stated that in this case we can do so since the fundamental group $\Lambda$ of $X\backslash \text{Sing}(X)$ is the group $\mathbb{Z}/4\mathbb{Z} \ltimes (\mathbb{Z}[i]\times \mathbb{Z}[i])$. They further say that $\mathbb{Z}[i]\times \mathbb{Z}[i]$ is invariant under all automorphism of $\Lambda$ and used this to conclude that we can lift an aut on $X$ to that of $Y$.

How did they arrive at this conclusion? Would be grateful if anyone can help to explain what that paragraph meant.