  We study the 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. In the complement of
this set, we found 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 ${\rm SL}(2,\mathbb{R})-$cocycles, and Jager-Wang-Zhou's result on
quasi-periodically forced maps, to a broader setting.
