The uniform word problem for groups and finite Rees quotients of E-unitary inverse semigroups

If is a class of groups closed under taking subgroups, we show that the decidability of the uniform word problem for is implied by the decidability of the membership problem for the class of finite Rees quotients of E-unitary inverse semigroups with maximal group image in . The converse is shown if is a pseudovariety. When is a pseudovariety, the above problems are shown to be equivalent to the problem of embedding a finite labeled graph in the Cayley graph of a group in . This latter problem is shown to be equivalent to deciding whether a finite labeled graph is a Schützenberger graph of an E-unitary inverse semigroup with maximal group image in .