  In this paper, we present a generalization of Hayden's theorem [7, Theorem
4.2] for $G$-codes over finite Frobenius rings. A lattice theoretical form of
this generalization is also given. Moreover, Astumi's MacWilliams identity [1,
Theorem 1] is generalized in several ways for different weight enumerators of
$G$-codes over finite Frobenius rings. Furthermore, we provide the Jacobi
analogue of Astumi's MacWilliams identity for $G$-codes over finite Frobenius
rings. Finally, we study the relation between $G$-codes and its corresponding
$G$-lattices.
