Let S be a finite semigroup,
Abe a given subset of S and L, R, H, Dand Jbe Green's equivalence relations. The problem of determining whether there exists a supersemigroup T of S from the class of all semigroups or from the class of finite semigroups, such that Alies in an Lor R-class of T has a simple and well known solution (see for example ,  or ). The analogous problem for Jor Drelations is trivial if T is of arbitrary size, but undecidable if T is required to be finite  (even if we restrict ourselves to the case | A| = 2 ). We show that for the relation H, the corresponding problem is undecidable in both the class of finite semigroups (answering Problem 1 of ) and in the class of all semigroups, extending related results obtained by M. V. Sapir in . An infinite semigroup with a subset never lying in a H-class of any embedding semigroup is known and, in , the existence of a finite semigroup with this property is established. We present two eight element examples of such semigroups as well as other examples satisfying related properties.