Representation stability for homotopy automorphisms

We consider in parallel pointed homotopy automorphisms of iterated wedge sums of finite CW–complexes and boundary-relative homotopy automorphisms of iterated connected sums of manifolds minus a disk. Under certain conditions on the spaces and manifolds, we prove that the rational homotopy groups of these homotopy automorphisms form finitely generated FI–modules, and thus satisfy representation stability for symmetric groups in the sense of Church and Farb. We also calculate explicit bounds on the weights and stability degrees of these FI–modules.