let $A\ncong \mathbb Z_2 , \{e\}$ abelian group, i want to find a automorphism $\varphi\neq Id_A$.
i tried to define it such that for every $a\in A $ , $\varphi (a)=-a$. this definition will do unless for every $a\in A$ we will find that $a^2=e$. now here i tied to think maybe it can to be a vector space over $\mathbb Z_2$ and then to use maybe a permutation? we i dont know. i will be happy for some help.
You are almost done. Denote $A$ with an additive notation. If $\forall x \in A, x+x=0$, then it is well defined the $\mathbb{Z}_2$-vector space structure on $A$ $$\lambda x = \left\{ \begin{matrix} x & \mbox{ if } & \lambda = 1\\ 0 & \mbox{ if } & \lambda = 0 \end{matrix} \right. $$ for $\lambda \in \mathbb{Z}_2, x \in A$.
Since $A \ncong \mathbb{Z}_2$, the dimension of $A$ must be at least $2$. Let $B$ a basis of $A$, $\sigma$ a permutation of $B$ which is not $id_B$. Then $\sigma$ induces a non-identical automorphism $f \in \operatorname{Aut} (A)$ as a $\mathbb{Z}_2$-vector space.
And clearly this is a non-identical automorphism of $A$ as a group.