Bott inverted mod $l$ algebraic $K$-theory coincides with mod $l$ etale $K$-theory under mild conditions. The Bott element is an element in $K_2$ that is inverted with respect to the graded product structure on the algebraic $K$-theory. How does $K_n(X,\mathbb{Z}/l)$ compare to $K[\beta^{-1}]_n(X,\mathbb{Z}/l)$? Does one necessarily inject into the other one? If $K_n(X,\mathbb{Z}/l)$ is non-zero is $K[\beta^{-1}]_n(X,\mathbb{Z}/l)$ necessarily non-zero or can inverting the Bott element kill everything in the group?
So the question is whether the Bott element mod $l$ can be nilpotent or not? (If not the localization is going to be injective if it is nilpotent then $K[\beta^{-1}]_n(X,\mathbb{Z}/l)=0$ for $n\geq 2$.)