A family of pseudovarieties of solvable groups is constructed, each of which has decidable membership and undecidable extension problem for partial permutations. Included are a pseudovariety U satisfying no non-trivial group identity and a metabelian pseudovariety Q. For each of these pseudovarieties V, the inverse monoid pseudovariety Sl * V has undecidable membership problem. As a consequence, it is proved that the pseudovariety operators * , * * , m , ♦, ♦n, and P do not preserve

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... . In addition, several joins, including A ∨ U, are shown to be undecidable.