Generalized Burnside-Grothendieck ring functor and aperiodic ring functor associated with profinite groups

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.