Let $G$ be a finite group acting on a vector space $V = \mathbb{ F}_p^n$ over a prime field. Given finite sets $S \subset G$ and $E \subset V$, we study the restricted orbit union $S(E) = \bigcup_{ g\in S} g(E)$ and establish quantitative lower bounds for $|S(E)|$ in terms of $|S|$, $|E|$, and natural structural conditions. This finite field packing problem has connections to distance geometry, configuration counting, and expanding graphs. For $G = SL_2(\mathbb{ F}_p)$ acting on $\mathbb{ F}_p^2$, we prove that $$|S(E)| \gg \min\left\lbrace p^2, \frac{ |S||E|}{ p^2}\right\rbrace,$$ which is sharp. Under geometric non-concentration conditions on $E$ and subgroup-avoidance hypotheses on $S$, we obtain a power-saving improvement of the form $$|S(E)|\gg \min \left\lbrace p^2, ~\max\left\lbrace\frac{ |S||E|}{ pk}, ~\frac{ |S|^{ \frac{ 1}{ 2}}|E|}{ p^{ \frac{ 1-\epsilon}{ 2}}k^{ \frac{ 1}{ 2}}}\right\rbrace \right\rbrace,$$ where $k$ bounds the radial multiplicity of $E$. For small sets $|E| \leq p$, we establish optimal bounds using weighted incidence theory. Analogous results are proved for the first Heisenberg group $\mathbb{ H}_1(\mathbb{ F}_p)$ acting on $\mathbb{ F}_p^3$. Our approach reformulates the problem as an incidence question in a bipartite action graph. The proofs combine Fourier analytic techniques, energy estimates, point-line incidence bounds, and area-energy inequalities for skew dot products. The methods extend classical sum-product type problems and incidence theory to noncommutative group actions.
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