We obtain sharp results for the gencricity and stability of transitivity, ergodicity and mixing for compact connected Lie group extensions over a hyperbolic basic set of a C-2 diffeomorphism. In contrast to previous work, our results hold for general hyperbolic basic sets and, are valid in the C-r-topology for all r > 0 (here r need not be an integer and C-1 is replaced by Lipschitz). Moreover, when r >= 2, we show that there is a C-2-open and C-r-dense subset of C-r -extensions that are ergodic. We obtain similar results on stable transitivity for (non-compact) R-m-extensions, thereby generalizing a result of Nitica and Pollicott, and on stable mixing for suspension flows.