We extend recent results of
Adler, Kitchens & Shub, Parry, and Parry & Pollicott
on the stable ergodicity and mixing of toral extensions to skew
extensions by compact connected Lie groups. We show that
Holder continuous extensions by a compact
Lie group over a hyperbolic attractor are generically
stably ergodic. If the group is compact semisimple, then
Holder continuous extensions over hyperbolic sets are generically
stably ergodic.