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

We show that the rational subset membership problem in G can be reduced to the submonoid membership problem in G×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.