Amenable uniformly recurrent subgroups and lattice embeddings

We study lattice embeddings for the class of countable groups $\mathrm{\Gamma }$ defined by the property that the largest amenable uniformly recurrent subgroup $A_{\mathrm{\Gamma }}$ is continuous. When $A_{\mathrm{\Gamma }}$ comes from an extremely proximal action and the envelope of $A_{\mathrm{\Gamma }}$ is co-amenable in $\mathrm{\Gamma }$, we obtain restrictions on the locally compact groups $G$ that contain a copy of $\mathrm{\Gamma }$ as a lattice, notably regarding normal subgroups of $G$, product decompositions of $G$, and more generally dense mappings from $G$ to a product of locally compact groups.