  The excitation of quadratic quasinormal modes is an important nonlinear
phenomenon for a Kerr black hole ringing at a specific linear mode. The
amplitude of this second-order effect is proportional to the square of the
linear mode amplitude, with the ratio being linked to the nature of the Kerr
black hole. Focusing on the linear $(l=m=2,n=0)$ mode, we compute the
dependency of the ratio on the dimensionless spin of the black hole, ranging up
to 0.99, with the method applicable for more general mode couplings. Our
calculation makes use of the frequency-domain, second-order Teukolsky equation,
which involves two essential steps (a) analytically reconstructing the metric
through the Chrzanowski-Cohen-Kegeles approach and (b) numerically solving the
second-order Teukolsky equation using the shooting method along a complex
contour. We find that the spin dependence of the ratio shows a strong
correlation with the angular overlap between parent and child modes, providing
qualitative insights into the origin of the dependence. Depending on the nature
of the angular overlap, the ratio decreases with spin in scenarios such as the
channel $(l=m=2,n=0)\times(l=m=2,n=0)\to(l=m=4)$ or increases in situations
like $(l=m=2,n=0)\times(l=m=2,n=0)\to(l=5,m=4)$. For both cases, the ratios do
not vanish in the extremal limit. As a byproduct, we find that the Weyl scalars
can be concisely expressed with the Hertz potential.
