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In difference to many recent articles that deal with generalized bent (gbent)functions $f:\mathbb{Z}_2^n \rightarrow \mathbb{Z}_q$ for certain small valued$q\in \{4,8,16 \}$, we give a complete description of these functions for both$n$ even and odd and for any $q=2^k$ in terms of both the necessary andsufficient conditions their component functions need to satisfy. This enablesus to completely characterize gbent functions as algebraic objects, namely asaffine spaces of bent or semi-bent functions with interesting additionalproperties, which we in detail describe. We also specify the dual and the Grayimage of gbent functions for $q=2^k$. We discuss the subclass of gbentfunctions which corresponds to relative difference sets, which we call$\mathbb{Z}_q$-bent functions, and point out that they correspond to a class ofvectorial bent functions. The property of being $\mathbb{Z}_q$-bent is muchstronger than the standard concept of a gbent function. We analyse two examplesof this class of functions.

Full characterization of generalized bent functions as (semi)-bent spaces, their dual, and the Gray image