  In Pasalic et al., IEEE Trans. Inform. Theory 69 (2023), 2702--2712, and in
Anbar, Meidl, Cryptogr. Commun. 10 (2018), 235--249, two different vectorial
negabent and vectorial bent-negabent concepts are introduced, which leads to
seemingly contradictory results. One of the main motivations for this article
is to clarify the differences and similarities between these two concepts.
Moreover, the negabent concept is extended to generalized Boolean functions
from \(\mathbb{F}_2^n\) to the cyclic group \(\mathbb{Z}_{2^k}\). It is shown
how to obtain nega-\(\mathbb{Z}_{2^k}\)-bent functions from
\(\mathbb{Z}_{2^k}\)-bent functions, or equivalently, corresponding
non-splitting relative difference sets from the splitting relative difference
sets. This generalizes the shifting results for Boolean bent and negabent
functions. We finally point to constructions of \(\mathbb{Z}_8\)-bent functions
employing permutations with the \((\mathcal{A}_m)\) property, and more
generally we show that the inverse permutation gives rise to
\(\mathbb{Z}_{2^k}\)-bent functions.
