Can someone concisely list all characteristic classes (i.e., the cohomology classes $H^*(BX,A)$ of the corresponding classifying spaces) for the most relevant structure groups $X$ such as $O(n)$, $SO(n)$, $U(n)$, $SU(n)$ and finite abelian groups $A$ such as $\mathbb{Z}_2$ and $\mathbb{Z}$, and others if you feel like? Additionally, the relations between the different classes would be interesting, such as restriction of the $\mathbb{Z}$-classes to $\mathbb{Z}_2$ classes.
I know that the $\mathbb{Z}_2$-valued classes for $O(n)$ are polynomials of Stiefel-Whitney classes, but I'm having trouble finding the results for $\mathbb{Z}$-valued classes. Anyways, the results seem quite scattered, and I think it would be good if there was an easy-access place where all (or at least the most important ones of) the classes are summarized. If such a place exists already, I'd be happy with a link of course.