A major result of D. B. McAlister is that every inverse semigroup is an idempotent separating morphic image of an E-unitary inverse semigroup. The result has been generalized by various authors (including Szendrei, Takizawa, Trotter, Fountain, Almeida, Pin, Weil) to any semigroup of the following types: orthodox, regular, ii-dense with commuting idempotents, E-dense with idempotents forming a subsemigroup, and is-dense. In each case, a semigroup is a morphic image of a semigroup in which the weakly self conjugate core is unitary and separated by the homomorphism. In the present paper, for any variety H of groups and any E-dense semigroup S, the concept of an “H-verbal subsemigroup” of S is introduced which is intimately connected with the least H-congruence on S. What is more, this construction provides a short and easy access to covering results of the aforementioned kind. Moreover, the results are generalized, in that covers over arbitrary group varieties are constructed for any E-dense semigroup. If the given semigroup enjoys a “regularity condition” such as being eventually regular, group bound, or regular, then so does the cover.