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 (X-H)H <= G for a G-space X, together with a pointed homotopy action of the group NGH/H on X-H/(U-H<K X-K). We then describe a procedure for constructing a suitable diagram <(X)under bar> : O-G(degrees P) -> 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 Bredort G-homotopy type. Such lifting methods may also be used to understand other homotopy questions about group actions, such as transferring a &action along a map f : X -> Y.