Comparison of K-groups of (affine) singular schemes with K'=G-groups

Question

It is well known that Quillen K-theory coincides with $K'=G$-theory for regular schemes, and can be distinct from it for singular ones. Are there any methods for studying this distinctions? In particular, for which $X$ we necessarily have $K_i(X)\cong K'_i(X)$ for $i\le n$ (for some $n\ge 0$), and how can one construct $X$ for which there is no such isomorphism? I am mostly interested in affine noetherian integral schemes over a field (of characteristic $\neq l$; the case of schemes over complex numbers is quite interesting) and in $K$-theory with $\mathbb{Z}/l\mathbb{Z}$-coefficients. Any hints or references would be very welcome!