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For a variety of finite groups $\mathbf H$, let $\overline{\mathbf H}$ denotethe variety of finite semigroups all of whose subgroups lie in $\mathbf H$. Wegive a characterization of the subsets of a finite semigroup that are pointlikewith respect to $\overline{\mathbf H}$. Our characterization is effectivewhenever $\mathbf H$ has a decidable membership problem. In particular, theseparation problem for $\overline{\mathbf H}$-languages is decidable for anydecidable variety of finite groups $\mathbf H$. This generalizes Henckell'stheorem on decidability of aperiodic pointlikes.

Pointlike sets for varieties determined by groups