Isovariant extensors and the characterization of equivariant homotopy equivalences

We extend the well-known theorem of James–Segal to the case of an arbitrary family $\mathcal{F}$ of conjugacy classes of closed subgroups of a compact Lie group $G$: a $G$-map $f\colon\mathbb{X}\to\mathbb{Y}$ of metric $\operatorname{Equiv}_{\mathcal{F}}$-$\mathrm{ANE}$-spaces is a $G$-homotopy equivalence if and only if it is a weak $G$-$\mathcal{F}$-homotopy equivalence. The proof is based on the theory of isovariant extensors, which is developed in this paper and enables us to endow $\mathcal{F}$-classifying $G$-spaces with an additional structure.