${\rm Aut}(G)$ is cyclic $\implies G$ is abelian

There is a nice chain of small results which proves this which continues down the path that Groups suggests. If ${\rm Aut}(G)$ is cyclic, then so is any subgroup of it, in particular ${\rm Inn}(G)$. ${\rm Inn}(G)\cong G/Z(G)$ where $Z(G)$ is the center. If $G/Z(G)$ is cyclic, the group is abelian.

The $\phi,\psi$ commute, and also the following steps are also OK: $$\phi\psi(g_1g_1)=\phi\psi(g_1)\phi\psi(g_2)=g_3g_4.$$ Its not clear in your argument why $g_3g_4=\phi\psi(g_2)\phi\psi(g_1)$?

We are allowed to use commutativity of maps $\phi,\psi$, and we have to conclude commutativity of $g_1,g_2$. You may proceed in following directions.

(1) Consider a specific subgroup of ${\rm Aut}(G)$, namely ${\rm Inn}(G)$. How it is related with $G$?

(2) ${\rm Aut}(G)$ is cyclic, so is ${\rm Inn}(G)$, then using (1), what this will imply?