$\newcommand{\Id}{\operatorname{Id}}$
$f$ is an automorphism of an infinite cyclic group $G$ then
1.$f^n\neq \Id_G$
2.$f^2=\Id_G$
3.$f=\Id_G$
if $f^n=\Id_G$ then every element of $G$ will have finite order but in an infinite cyclic group only identity element has finite order, same for 2, so $3$ is correct?