  The extension of majorization (also called the rearrangement ordering), to
more general groups than the symmetric (permutation) group, is referred to as
$G$-majorization. There are strong results in the case that $G$ is a reflection
group and this paper builds on this theory in the direction of subgroups,
normal subgroups, quotient groups and extensions. The implications for
fundamental cones and order-preserving functions are studied. The main example
considered is the hyperoctahedral group, which, acting on a vector in $\mathbb
R^n$, permutes and changes the signs of components.
