  Let $G$ and $H$ be finite groups. We give a condition on $G$ and $H$ that
implies that the $A$-fibered Burnside rings $B^A(G)$ and $B^A(H)$ are
isomorphic. As a consequence, we show the existence of non-isomorphic groups
$G$ and $H$ such that $B^A(G)$ and $B^A(H)$ are isomorphic rings. Here, the
abelian fiber group $A$ can be chosen in a non-trivial way, that is, such that
$B^A(G)$ and $B^A(H)$ are strictly bigger than the Burnside rings of $G$ and
$H$, for which such counterexamples are already known.
