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The aim of this paper is to prove that a 
$\text{K3}$ surface is the minimal model of the quotient of an Abelian surface by a group 
$G$ (respectively of a $\text{K3}$ surface by an Abelian group $G$) if and only if a certain lattice is primitively embedded in its Néron-Severi group. This allows one to describe the coarse moduli space of the $\text{K3}$ surfaces that are (rationally) $G$-covered by Abelian or $\text{K3}$ surfaces (in the latter case $G$ is an Abelian group). When $G$ has order 2 or $G$ is cyclic and acts on an Abelian surface, this result is already known; we extend it to the other cases.
Moreover, we prove that a $\text{K3}$ surface 
${{X}_{G}}$ is the minimal model of the quotient of an Abelian surface by a group $G$ if and only if a certain configuration of rational curves is present on ${{X}_{G}}$. Again, this result was known only in some special cases, in particular, if $G$ has order 2 or 3.
