  For every profinite group $G$, we construct two covariant functors $\Delta_G$
and ${\bf {\mathcal {AP}}}_G$ from the category of commutative rings with
identity to itself, and show that indeed they are equivalent to the functor
$W_G$ introduced in [A. Dress and C. Siebeneicher, The Burnside ring of
profinite groups and the Witt vectors construction, {\it Adv. in Math.}
{\bf{70}} (1988), 87-132]. We call $\Delta_G$ the generalized
Burnside-Grothendieck ring functor and ${\bf {\mathcal {AP}}}_G$ the aperiodic
ring functor (associated with $G$). In case $G$ is abelian, we also construct
another functor ${\bf Ap}_G$ from the category of commutative rings with
identity to itself as a generalization of the functor ${\bf Ap}$ introduced in
[K. Varadarajan, K. Wehrhahn, Aperiodic rings, necklace rings, and Witt
vectors, {\it Adv. in Math.} {\bf 81} (1990), 1-29]. Finally it is shown that
there exist $q$-analogues of these functors (i.e, $W_G, \Delta_G, {\bf
{\mathcal {AP}}}_G$, and ${\bf Ap}_G$) in case $G=\hat C$ the profinite
completion of the multiplicative infinite cyclic group.
