I know that the general word problem is undecidable, but is there an effective enumeration of presentations all finitely presented groups generated by $n$ elements in which each isomorphism class of a group appears exactly once?
I have no idea of how to tackle this problem, and I'm motivated by the fact that isomorphism classes of path connected covering maps $E \to \bigvee_n S^1$ correspond precisely to subgroups of the free group of $n$ elements. So answering the previous question solves the one about covering maps too.
Since the isomorphism problem for finitely presented groups is undecidable, there can be no such enumeration, at least for $n \ge 2$.
If there were such an enumeration $G_1,G_2,\ldots,G_n,\ldots$ then I could solve the isomorphism problem as follows. Given a finite presentation on $n$ generators, attempt to construct isomorphisms between the group defined by this presentation and the groups $G_1,G_2,\ldots,G_n,\ldots$ in parallel. Eventually an isomorphism with exactly one of the groups $G_i$ would be found.