This problem come from next question:
If finite group $G$ have non fixed point automorphism $\alpha$ with $\alpha^2 = Id_G$,prove G is a Ablian group with odds order.
There:$a\ne 1\in G$ of group morphism $\alpha$ is a fixed point if $\alpha(a)=a$.
I Think this $\alpha$ is $$\alpha:G\to G\\ g\to g^{-1}$$ Firstly ,I want to use the non fixed point property .Then ,for any $g\in G$,we conserder $$\alpha(\alpha(g)g)=g\alpha(g)$$ If $g\alpha(g)=\alpha(g)g$,this problem is closed.But I think it's not clearly to prove $g\alpha(g)=\alpha(g)g$.How can you do?