A closed subgroup $H$ of a connected reductive group $G$ is called spherical if a Borel subgroup of $G$ has an open orbit in the homogeneous space $G/H$. According to a result of Vinberg and Kimelfeld from 1978, $H$ is spherical if and only if for every irreducible representation $V$ of $G$ and every one-dimensional representation $W$ of $H$, the multiplicity of $W$ in $V$ is at most one. All possible pairs $(V,W)$ for which the multiplicity equals one are encoded in a semigroup, called the extended weight semigroup of $G/H$, and a natural problem is to compute this semigroup for any given spherical subgroup $H$. The extended weight semigroups are known for several classes of spherical subgroups including reductive and solvable ones. The goal of the talk is to discuss possible approaches for computing this semigroup for arbitrary subgroups.