Let $G$ be a group. The automorphism group $\mathrm{Aut}(G)$ acts on $G$ by $\psi\cdot g=\psi(g)$ for all $\psi\in\mathrm{Aut}(G)$ and all $g\in G$. Find the orbits of this action for $G=\mathbb Z_{12}$, $\mathbb Z_7$ and $D_4$.
In cases for first two, since $\mathrm{Aut}(Z_n)\cong U(n)$, by computation, the followings are orbits:
For $\mathbb Z_{12}$,
$$\mathrm{Aut}(G)\cdot0 =\{0\}$$
$$\mathrm{Aut}(G)\cdot1 =\{1,5,7,11\}$$
$$\mathrm{Aut}(G)\cdot2 =\{2,10\}$$
$$\mathrm{Aut}(G)\cdot3 =\{3,9\}$$
$$\mathrm{Aut}(G)\cdot4 =\{4,6,8\}$$
$$\mathrm{Aut}(G)\cdot6 =\{6\}$$
$$\mathrm{Aut}(G)\cdot8 =\{4,8\}$$
For $\mathbb Z_7$,
$$\mathrm{Aut}(G)\cdot0 =\{0\}$$ $$\mathrm{Aut}(G)\cdot1 =\{1,2,3,4,5,6\}$$
But, in case of $D_4$, dihedral group of order 8, I don't know what the automorphism group is. And this is an exercise for groups acting on themselves such as left multiplication and conjugation. How could I use it to solve this problem?
