A cryptogroup \(S\) is a completely regular semigroup in which the Green relation \(\mathcal H\) is a congruence, that is a band of groups. If, in addition, \(S/{\mathcal H}\) is a normal band, then \(S\) is a normal cryptogroup or a normal band of groups. A semilattice of groups \(G=[Y;G_\alpha,w_{\alpha,\beta}]\) is called a Clifford semigroup. Given a Clifford semigroup \(G\), the author constructs special \(G\)-operands \(L\) and \(R\). Certain suboperands of \(L\) and \(R\) are called threads. A new representation of completely simple semigroups is handled; a very special \(G\)-operand, where \(G\) is a Clifford semigroup, is constructed; the basic concept of a coherent \(G\)-isomorphism of threads is introduced; a correspondence between the semigroups of coherent \(G\)-isomorphisms of cyclic operands and the class of normal cryptogroups satisfying a special condition is established and this is extended to include the semigroups of all coherent \(G\)-isomorphisms of threads and the semigroups of threads of normal cryptogroups satisfying a special condition; several sufficient conditions on normal cryptogroups in order that there exist representations satisfying a special condition are given.