Recently in Çeşmelioğlu, Meidl (Adv. Math. Commun., 18, 2024), the study of EA-equivalence and CCZ-equivalence for functions from \({\mathbb {V}}_n^{(p)}\) to the cyclic group \({\mathbb {Z}}_{p^k}\) has been initiated, where \( {\mathbb {V}}_n^{(p)}\) denotes an n-dimensional vector space over \({\mathbb {F}}_p\) . Amongst others it has been shown that there exist functions from \({\mathbb {V}}_n^{(2)}\) to \({\mathbb {Z}}_4\) which are CCZ-equivalent but not EA-equivalent. We extend these results to larger classes of functions from \({\mathbb {V}}_n^{(p)}\) to \({\mathbb {Z}}_{p^k}\) . We then discuss constructions of generalized bent functions from \({\mathbb {V}}_n^{(p)}\) to \({\mathbb {Z}}_{p^k}\) , p odd or \(p=2\) and n is even, which correspond to large affine spaces of bent functions. In particular we employ versions of the direct sum, the semi-direct sum and of a recent secondary bent function construction in Wang et. al., (IEEE Trans. Inform. Theory 69, 2023), to generate large affine spaces of bent functions. Finally we present a solution for constructing generalized bent functions from \({\mathbb {V}}_n^{(2)}\) to \({\mathbb {Z}}_{2^k}\) , n odd, from arbitrary generalized bent functions from \({\mathbb {V}}_{n-1}^{(2)}\) to \({\mathbb {Z}}_{2^{k-1}}\) .
