  The quotient space of a $K3$ surface by a finite group is an Enriques surface
or a rational surface if it is smooth. Finite groups where the quotient space
are Enriques surfaces are known. In this paper, by analyzing effective divisors
on smooth rational surfaces, we will study finite groups which act faithfully
on $K3$ surfaces such that the quotient space are smooth. In particular, we
will completely determine effective divisors on Hirzebruch surfaces such that
there is a finite Abelian cover from a $K3$ surface to a Hirzebrunch surface
such that the branch divisor is that effective divisor. Furthermore, we will
decide the Galois group and give the way to construct that Abelian cover from
an effective divisor on a Hirzebruch surface. Subsequently, we study the same
theme for Enriques surfaces.
