Amenability constants of central Fourier algebras of finite groups

We consider amenability constants of the central Fourier algebra $ZA(G)$ of a finite group $G$. This is a dual object to $ZL^1(G)$ in the sense of hypergroup algebras, and as such shares similar amenability theory. We provide several classes of groups where $\mathrm{AM}(ZA(G)) = \mathrm{AM}(ZL^1(G))$, and discuss $\mathrm{AM}({ZA}(G))$ when $G$ has two conjugacy class sizes. We also produce a new counterexample which shows that unlike $\mathrm{AM}({ZL}^1(G))$, $\mathrm{AM}({ZA}(G))$ does not respect quotient groups, but the class of groups that it does has $\frac {7}{4}$ as the sharp amenability constant bound.