Automorphism Orbits of the Group of Unitriangular Matrices

Let $G$ be a group. The orbits of the natural action of $Aut(G)$ on $G$ are called the automorphism orbits of $G$, and their number is denoted by $\omega (G)$. Let $\mathbb{F}$ be an infinite field, and let $U{T}_n(\mathbb{F})$ denote the group of unitriangular matrices over $\mathbb{F}$. We show that $\omega (U{T}_n(\mathbb{F}))$ is finite for $n\le 5$ and infinite for $n\ge 6$.