  Cocycles are a key object in Antol\'{i}n Camarena and Szegedy's (topological)
theory of nilspaces. We introduce measurable counterparts, named nilcycles,
enabling us to give conditions which guarantee that an ergodic group extension
of a strictly ergodic distal system admits a strictly ergodic distal
topological model, revisiting a problem studied by Lindenstrauss. In particular
we show that if the base space is a dynamical nilspace then a dynamical
nilspace topological model may be chosen for the extension. This approach
combined with a structure theorem of Gutman, Manners and Varj\'{u} applied to
the ergodic group extensions between successive Host-Kra characteristic factors
gives a new proof that these factors are inverse limit of nilsystems.
