  Let $A$ be an abelian group such that $\mathrm{Hom}(G,A)$ is finite for all
finite groups $G$, and let $\mathbb{K}$ be a field of characteristic zero
containing roots of unity of all orders equal to finite element orders in $A$.
In this paper we prove foundational properties of the $A$-fibered Burnside ring
functor $B_{\mathbb{K}}^A$ as an $A$-fibered biset functor over $\mathbb{K}$.
This includes the determination of the lattice of subfunctors of
$B_{\mathbb{K}}^A$ and the determination of the composition factors of
$B_{\mathbb{K}}^A$. The results of the paper extend results of Co\c{s}kun and
Y\i lmaz for the $A$-fibered Burnside ring functor restricted to $p$-groups and
results of Bouc in the case that $A$ is trivial, i.e., the case of the Burnside
ring functor over fields of characteristic zero.
