Enriques surfaces and Leech lattice

Let $L$ be an even unimodular lattice of signature $(1,25)$ which is unique up to isomorphisms. J.H. Conway found a fundamental domain $C$ of the reflection group of $L$ by using a theory of Leech lattice. Recently S. Brandhorst and I. Shimada have classified all primitive embeddings of $E_{10}(2)$ into $L$, where $E_{10}(2)$ is the pullback of the Picard lattice of an Enriques surface to the covering K3 surface. There are exactly $17$ embeddings. By restricting $C$ to the positive cone of $E_{10} \otimes \mathbf{R}$ we obtain $17$ polyhedrons. In this talk I would like to discuss the automorphism groups of Enriques and Coble surfaces in terms of these polyhedrons.