We study circle homeomorphisms extensions over a strictly ergodic homeomorphism. Under a very mild restriction, we show that the fibered rotation number is locally constant on an open and dense subset of all circle home-omorphisms extensions homotopic to the trivial extension. In the complement of this set, we find a dense subset in which every map is conjugate to a direct product. Our result provides a generalisation of Avila-Bochi-Damanik's result on SL(2, R)−cocycles, and Jäger-Wang-Zhou's result on quasi-periodically forced maps, to a broader setting.