Classical gravitational scattering amplitude at <math display="inline"><mi mathvariant="script">O</mi><mo stretchy="false">(</mo><msup><mi>G</mi><mn>2</mn></msup><msubsup><mi>S</mi><mn>1</mn><mi>∞</mi></msubsup><msubsup><mi>S</mi><mn>2</mn><mi>∞</mi></msubsup><mo stretchy="false">)</mo></math>

We calculate the scattering amplitude of two rotating objects with the linear-in-curvature spin-induced multipoles of Kerr black holes at $\mathcal{O}(G^2)$ and all orders in the spins of both objects. This is done including the complete set of contact terms potentially relevant to Kerr-black-hole scattering at $\mathcal{O}(G^2)$. As such, Kerr black holes should be described by this scattering amplitude for a specific choice of values for the contact-term coefficients. The inclusion of all potential contact terms means this amplitude allows for a comprehensive search for structures emerging for certain values of the coefficients, and hence special properties that might be exhibited by Kerr-black-hole scattering. Our result can also act as a template for comparison for future computations of classical gravitational high-spin scattering.