Measures induced by sugroups and tuples in free groups

We study probability measure on $\mathrm{Hom}(H,G)$, where $G$ is a finite group and $H$ a finitely generated subgroup of a finitely generated free group $F$, obtained by pushing forward the uniform random homomorphisms $\mathrm{Hom}(F,G)$ via restriction map to $\mathrm{Hom}(H,G)$. This framework generalizes the word measures arising from single elements of a free group. We formalize the notion of profinite rigidity for subgroups via these induced measures. Our main result shows that a finitely generated subgroup is profinitely rigid if and only if any (equivalently, every) ordered generating tuple is profinitely rigid, thereby extending the notion of rigidity from individual word maps to arbitrary tuples. We also obtain a generalization of a result of \cite{puder2015measure}.