  For $G$ a finite abelian group, we study the properties of general
equivalence relations on $G_n=G^n\rtimes \SG_n$, the wreath product of $G$ with
the symmetric group $\SG_n$, also known as the $G$-coloured symmetric group. We
show that under certain conditions, some equivalence relations give rise to
subalgebras of $\k G_n$ as well as graded connected Hopf subalgebras of
$\bigoplus_{n\ge o} \k G_n$. In particular we construct a $G$-coloured peak
subalgebra of the Mantaci-Reutenauer algebra (or $G$-coloured descent algebra).
We show that the direct sum of the $G$-coloured peak algebras is a Hopf
algebra. We also have similar results for a $G$-colouring of the Loday-Ronco
Hopf algebras of planar binary trees. For many of the equivalence relations
under study, we obtain a functor from the category of finite abelian groups to
the category of graded connected Hopf algebras. We end our investigation by
describing a Hopf endomorphism of the $G$-coloured descent Hopf algebra whose
image is the $G$-coloured peak Hopf algebra. We outline a theory of
combinatorial $G$-coloured Hopf algebra for which the $G$-coloured
quasi-symmetric Hopf algebra and the graded dual to the $G$-coloured peak Hopf
algebra are central objects.
