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_={α₀,α₁,...,α_m-1} be a basis of GR(4^m) over Z₄ when we regard GR(4^m) as a free Z₄-module of rank m. Define the map d_{α_} from GR(4^m)[z]/(zⁿ-1) into Z₄[z]/(z^mn-1) by dα_(a(z))=Σ_i=0^m-1Σ_j=0^n-1a_ijz^mj+i where a(z)=Σ_j=0^n-1a_jz^j and a_j=Σ_i=0^m-1a_ijα_i, a_ij∈Z₄. Then, for any linear code C of length n over GR(4^m), its image d_{α_}(C) is a Z₄-linear code of length mn. In this article, for n and m being odd integers, it is determined all pairs (α_,C) such that d_{α_}(C) is Z₄-cyclic, where α_ is a basis of GR(4^m) over Z₄, and C is a cyclic code of length n over GR(4^m).