Normalizers of Chevalley groups of type $G_2$ over local rings without $1/2$

In this paper, we prove that every element of the linear group $\mathrm{GL}_{14} (R)$ normalizing the Chevalley group of type $G_2$ over a commutative local ring $R$ without $1/2$ belongs to this group up to some multiplier. This allows us to improve our classification of automorphisms of these Chevalley groups showing that an automorphism-conjugation can be replaced by an inner automorphism. Therefore, it is proved that every automorphism of a Chevalley group of type $G_2$ over a local ring without $1/2$ is a composition of a ring and an inner automorphisms.