This is a question from chapter $4D$ of Isaacs' Finite Group Theory.
Let $A$ act via automorphisms on $G$, where $G$ is a $2$-group and $A$ has odd order. Show that if $A$ fixes every element $x$ in $G$ such that $x^4=1$, then the action of $A$ on $G$ is trivial.
We get $A$ acts coprimely on $G$. The problem is I do not understand why $G$ is $2$-group and how I can use it. I think we should assume $A$ does not act trivially on $G$, then there exist an element $g$ in $G$ which is not fixed by $A$. Then we need to show $g^4\neq1$. How can I get this $g$ ?