Finite power image families and automorphism actions in linear groups

Let $k\mathrm{/}\mathbb{Q}$ be a finitely generated field extension. We characterize the subgroups $G\le G{L}_n(k)$ for which the power images $P_m(G)=\{g^m:g\in G\}$ form a finite family: this holds precisely when $G=N⋊F$, where $F$ is finite and $N$ is a uniquely divisible subgroup consisting of unipotent matrices. The subgroup $N$ is canonical and fully invariant.For $e=\exp (G\mathrm{/}N)$, we obtain $P_m(G)=P_{\gcd (m,e)}(G)$, exactly $\tau (e)$ distinct power images, and least positive period $e$, where $\tau (e)$ is the number of positive divisors of $e$.Over $\mathbb{Q}$, the group coincides with the full group of rational points of its Zariski closure. Since finite automorphism-orbit count implies finiteness of the power-image family, an affirmative answer to Problem~21.40 of the Kourovka Notebook\cite{Kourovka} follows as a corollary. We further prove that a rational linear group $G$ admits a decomposition $G\cong ({\mathbb{Q}}^d,+)⋊F$, with $d\ge 0$ finite and $F$ finite, whenever a virtually nilpotent subgroup of $\mathrm{Aut}(G)$ has finitely many orbits on $G$.The latter argument uses nonconstant invariant rational functions arising from nonzero Lie brackets to obstruct dense orbits. A metabelian action with five orbits on the rational Heisenberg group shows that solubility of the acting group is insufficient.