Coverings of skew-products and crossed products by coactions

Consider a projective limit G of finite groups Gn. Fix a compatible family ⁿ of coactions of the Gn on a C^*-algebra A. From this data we obtain a coaction of G on A. We show that the coaction crossed product of A by is isomorphic to a direct limit of the coaction crossed products of A by the ⁿ. If A=C^*() for some k-graph , and if the coactions ⁿ correspond to skew-products of , then we can say more. We prove that the coaction crossed product of C^*() by may be realized as a full corner of the C^*-algebra of a (k+1)-graph. We then explore connections with Yeends topological higher-rank graphs and their C^*-algebras.