Invariant subalgebras rigidity for von Neumann algebras of groups arising as certain semidirect products

We study the ISR (von Neumann invariant subalgebra rigidity) property for certain discrete groups arising as semidirect products from algebraic actions on certain 2-torsion groups, mostly arising as direct products of ${\mathbb{Z}}_2$. We present, in particular, the first example of an amenable group with the ISR property that admits a non-trivial abelian normal subgroup. Several other examples are discussed, notably including an infinite amenable group whose von Neumann algebra admits precisely one invariant von Neumann subalgebra which does not come from a normal subgroup. We also investigate the form of invariant subalgebras of the group von Neumann algebra of the standard lamplighter group.