Linear stable ranges for integral homotopy groups of configuration spaces

We prove explicit linear stable ranges for the $\mathsf{F}\mathsf{I}$-modules $\mathrm{H}\mathrm{o}\mathrm{m}({\pi }_p\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{f}M,\mathbb{Z})$ and $\mathrm{E}\mathrm{x}\mathrm{t}({\pi }_p\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{f}M,\mathbb{Z})$ with $\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{f}M$ being the configuration co$\mathsf{F}\mathsf{I}$-space of a $d$-dimensional manifold with $d\ge 3$. The proof of this result uses a homotopy-theoretic approach to representation stability for $\mathsf{F}\mathsf{I}$-modules. This allows us to derive representation stability results from homotopy-theoretical statements, in particular the generalized Blakers-Massey theorem. We also generalize to ${\mathsf{F}\mathsf{I}}_G$-modules and orbit configuration spaces.