Realizing homotopy group actions

For any finite group G, we define the notion of a Bredon homotopy action of G, modelled on the diagram of fixed point sets (XH)H≤G for a G-space X, together with a pointed homotopy action of the group NGH/H on XH/(SH>K XK).We then describe a procedure for constructing a suitable diagram X : O op G → Top from this data, by solving a sequence of elementary lifting problems. If successful, we obtain a G-space X realizing the given homotopy information, determined up to Bredon G-homotopy type. Such lifting methods may also be used to understand other homotopy questions about group actions, such as transferring a G-Action along a map f : X → Y.