  The aim of this paper is to generalize results known for the symplectic
involutions on K3 surfaces to the order 3 symplectic automorphisms on K3
surfaces. In particular, we will explicitly describe the action induced on the
lattice $\Lambda_{K3}$, isometric to the second cohomology group of a K3
surface, by a symplectic automorphism of order 3; we exhibit the maps $\pi_*$
and $\pi^*$ induced in cohomology by the rational quotient map
$\pi:X\dashrightarrow Y$, where $X$ is a K3 surface admitting an order 3
symplectic automorphism $\sigma$ and $Y$ is the minimal resolution of the
quotient $X/\sigma$; we deduce the relation between the N\'eron--Severi group
of $X$ and the one of $Y$. Applying these results we describe explicit
geometric examples and generalize the Shioda--Inose structures, relating
Abelian surfaces admitting order 3 endomorphisms with certain specific K3
surfaces admitting particular order 3 symplectic automorphisms.
