On sumsets of nonbases of maximum size

Let $G$ be a finite abelian group. A nonempty subset $A$ in $G$ is called a basis of order $h$ if $hA=G$; when $hA\mathrm{\ne }G$, it is called a nonbasis of order $h$. Our interest is in all possible sizes of $hA$ when $A$ is a nonbasis of order $h$ in $G$ of maximum size; we provide the complete answer when $h=2$ or $h=3$.