The classes $KR_\infty$, $MR_\infty$, $R_\infty$ of rapidly varying functions are natural extensions of Karamata's concept of regular variation. In \cite{2} we introduced a new class $K$ of perfect Karamata's kernels and its subclasses $\Theta$ and $\Sigma$. In this paper we study inclusion properties of these classes and, among other results, we prove $KR_\infty\subset MR_\infty\subset\Sigma\subset\Theta\subset K$.