Cyclic codes over GR (4m) which are also cyclic over Z4

Let GR (4(m)) be,the Galois ring of characteristic 4 and cardinality 4(m), and (α) under bar = {alpha(0), alpha(1),..., alpha(m-1)} be a basis of GR (4(m)) over Z(4) when we regard GR (4(m)) as a free Z(4)-module of rank m. Define the map d((α) under bar) from GR (4(m)) [z] / (z(n) - 1) into Z(4) [Z] / (z(mn) - 1) by d((α) under bar)(a(z)) = Sigma(i=0)(m-1) Sigma(j=0)(n-1) a(ij)z(mj+i) where a(z) = Sigma(j=0)(n-1)a(j)z(j) and a(j) = Sigma(i=0)(m-1) a(ij)alpha(i), a(ij) is an element of Z(4). Then, for any linear code C of length n over GR (4(m)), its image d((α) under bar)(C) is a Z(4)-linear code of length mn. In this correspondence, for n and m being odd integers, it is determined all pairs ((α) under bar, C) such that d((α) under bar)(C) is Z(4)-cyclic, where (α) under bar is a basis of GR (4(m)) over Z(4), and C is a cyclic code of length n over GR (4(m)).