Birman–Hilden bundles. II

We study the structure of self-homeomorphism groups of fibered manifolds. A fibered topological space is a Birman–Hilden space whenever in each isotopic pair of its fiber-preserving (taking each fiber to a fiber) self-homeomorphisms the homeomorphisms are also fiber-isotopic (isotopic through fiber-preserving homeomorphisms). We prove in particular that the Birman–Hilden class contains all compact connected locally trivial surface bundles over the circle, including nonorientable ones and those with nonempty boundary, as well as all closed orientable Haken 3-manifold bundles over the circle, including nonorientable ones.