We obtain sharp results for the genericity and stability of transitivity, ergodicity and mixing for compact connected Lie group extensions over a hyperbolic basic set of a C2 diffeomorphism. In contrast to previous work, our results hold for general hyperbolic basic sets and are valid in the Cr-topology for all r > 0 (here r need not be an integer and C1 is replaced by Lipschitz). Moreover, when $r\ge2$, we show that there is a C2-open and Cr-dense subset of Cr-extensions that are ergodic. We obtain similar results on stable transitivity for (non-compact) $\mathbb{R}^m$-extensions, thereby generalizing a result of Niţică and Pollicott, and on stable mixing for suspension flows.