Automorphism group of a lattice's Voronoi cell
Solution 1:
The Voronoi cell $V$ determines the lattice $\Lambda$.
Consider an ($n-1$)-face $F$ of $V$. The hyperplane of the face $F$ is the perpendicular bisector of the segment between $O$ and a lattice point $p$. Then $V+p$ is the neighboring Voronoi cell that shares face $F$. Marching through the Voronoi cells we can find all cells and thus all lattice points at the centers of the cells.
Since every automorphism of $V$ maps $V$ to itself, they also map $\Lambda$ into itself. $Aut(V(\Lambda))\subset Aut(\Lambda)$.