Summary: We relate the problem of computing the closure of a finitely generated subgroup of the free group in the pro-\(\mathbf V\) topology, where \(\mathbf V\) is a pseudovariety of finite groups, with an extension problem for inverse automata which can be stated as follows: given partial one-to-one maps on a finite set, can they be extended into permutations generating a group in \(\mathbf V\)? The two problems are equivalent when \(\mathbf V\) is extension-closed. Turning to practical computations, we modify Ribes and Zalesskij’s algorithm to compute the pro-\(p\) closure of a finitely generated subgroup of the free group in polynomial time, and to effectively compute its pro-nilpotent closure. Finally, we apply our results to a problem in finite monoid theory, the membership problem in pseudovarieties of inverse monoids which are Mal’cev products of semilattices and a pseudovariety of groups.