  We establish exponential decay of correlations of all orders for locally
$G$-accessible isometric extensions of transitive Anosov flows, under the
assumption that the strong stable and strong unstable foliations of the base
Anosov flow are jointly $C^1$. This is accomplished by translating
accessibility properties of the extension into local non-integrability
estimates measured by Dolgopyat's infinitesimal transitivity group, from which
we obtain contraction properties for a class of 'twisted' symbolic transfer
operators.
