Abstract: Let S be a nil-extension of a Clifford semigroup K by a nil semigroup Q=S/K. By introducing a concept of admissible congruence pairs (δ,ω), where δ is a congruence on a nil semigroup Q and ω is a congruence on a Clifford semigroup K respectively, it is proved that every congruence σ on S can be uniquely represented by an admissible congruence pair on S. In addition, for any congruence σ on S, suppose that σ_K is a restriction of σ on a Clifford semigroup K, that is, σ_K=σ|_K and σ_Q=(σ∨ρ_K)/ρ_K, where ρ_K is a Rees congruence on S induced by a ideal K of S, it is proved that there is an order-preserving bijection Γ:σ→(σ_Q,σ_k) from the set of all congruences on S onto the set of all admissible congruence pairs on S. Finally, a condition has been given for a congruence which is a regular congruence on S.

Key words: admissible congruence pairs, C-quasiregular semigroups, nil-extensions