Membership problem for a subgroup

For a specified subgroup $H\le G$ and a chosen alphabet for $G$, the membership problem asks whether an input word represents an element of $H$. Solubility of the ordinary word problem for a group does not imply solubility of this problem, even for finitely generated subgroups: a Mihailova subgroup of a product of finite-rank free groups can have undecidable membership.