The original question was askes here.
I donot know how to apply or compute any example. I think a specified explanation will be helpful.
Let $M=\mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2$ and $f=A\in\mathrm{SL}(2,\mathbb{Z})$ be the quotient action on $\mathbb{T}^2$ induced from $f(x)=Ax$. Since $A$ preserves the lattice $\mathbb{Z}^2$, $f$ is well defined.
We know that $\pi_1(\mathbb{T}^2)=\langle\alpha,\beta\rangle\cong\mathbb{Z}^2$. What can we see about $\pi_1(M_f)$?
HJRW answered there that $\pi_1(M_f)\cong\pi_1(X)\rtimes_{f_*}\mathbb{Z}$. Would you write down explicitly the multiplication of this semiproduct in this example?
Thanks!