Is decidability of the Submonoid Membership Problem closed under finite extensions?

Abstract

We show that the rational subset membership problem in $G$ can be reduced to the submonoid membership problem in $G{\times}H$ where $H$ is virtually Abelian. We use this to show that there is no algorithm reducing submonoid membership to a finite index subgroup uniformly for all virtually nilpotent groups. We also provide evidence towards the existence of a group $G$ with a subgroup $H<G$ of index 2, such that the submonoid membership problem is decidable in $H$ but not in $G$.