Autocommutator subgroup $K(G)$ of a group $G$ is defined as $$K(G):=\langle [g,\alpha]:g\in G,\; \alpha \in Aut(G)\rangle .,$$
where $[g,\alpha ]:=g^{-1}\alpha (g)$. Is $K(G)$ a characteristic subgroup in $G$?
Recall that a subgroup $H$ of a group $G$ is said to be charactristic in $G$ if $\alpha (H)\subseteq H$ for all $\alpha \in Aut(G)$.
My try: Assume that $[g,\alpha ]\in K(G)$ is an arbitrary generator and $\beta \in Aut (G)$. We need to show that $\beta ([g,\alpha])\in K(G)$. We have $\beta ([g,\alpha])=\beta (g^{-1})\beta (\alpha (g))$. But I couldn't show that $\beta ([g,\alpha])\in K(G)$.