We consider the families of finite Abelian groups $\ZZ/p\ZZ\times \ZZ/q\ZZ$ and $\ZZ/p^2\ZZ$, for $p,q$ prime numbers. We give a simple characterization of all functions $f$ for which the size of the support is at most $k$ and the size of the spectrum is minimal among such functions. Such equality cases were previously known when $k$ divides the cardinal of the group, or for groups $\ZZ/p\ZZ$.