  A generalized Kummer surface $X$ of order $3$ is the minimal resolution of
the quotient of an abelian surface $A$ by an order $3$ symplectic automorphism.
We study a generalization of a problem of Shioda for classical Kummer surfaces,
which is to understand how much $X$ is determined by $A$ and conversely. The
surface $X$ posses a big and nef divisor $L_{X}$ such that $L_{X}^{2}=0$ or $2$
mod $6$. We show that for surfaces with $L_{X}^{2}=6k$ with $k\neq0,6\,mod\,9$,
the surface $X$ determines the transcendental lattice $T(A)$ of $A$ and the
Hodge structure on $T(A)$. Conversely if $A$ and $B$ are Fourier-Mukai partners
(i.e. if the Hodge structures of their transcendental lattices are isomorphic)
and $Y$ is the generalized Kummer surface which is the minimal resolution of
the quotient of $B$ by an order $3$ symplectic automorphism, we obtain that $X$
and $Y$ are isomorphic. These results are also know to hold for surfaces with
$L_{X}^{2}=2\,mod\,6$ from a previous work. When $k=0\text{ or }6\,mod\,9,$ we
show that $X$ determines $T(A)$ and its Hodge structure, but the converse does
not hold in general.
