Assume that $G$ is a finite group such that $|G| \gt 1$ and there exists $x \in G$ such that $x^2 \neq e$. Prove that there exists a non-trivial automorphism on $G$.
Note: This question has been asked and answered here. But, I can't understand that answers. Both of them use the concept of inner-automorphism. We haven't studied this in the class, So i'm not allowed to use the term 'Inner-Automorphism'. Also, I don't understand that what is their automorphism doing. We are given a group. Maybe its abelian. Maybe it's not. We should provide a function which takes an element of $G$ and maps it to another element of $G$. Which element is that another element?