Pointlike sets for varieties determined by groups

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For a variety of finite groups H, let H denote the variety of finite semigroups all of whose subgroups lie in H. We give a characterization of the subsets of a finite semigroup that are pointlike with respect to H. Our characterization is effective whenever H has a decidable membership problem. In particular, the separation problem for H-languages is decidable for any decidable variety of finite groups H. This generalizes Henckell's theorem on decidability of aperiodic pointlikes.