For n≥2, let Gn be a group and let ρ: Bn→Gn be a representation of the braid group Bn. For a field K and a,b,c∈K, Bardakov, Chbili, and Kozlovskaya extended the representation ρ to a family of representations Φa,b,c: SMn→K[Gn] of the singular braid monoid SMn, where K[Gn] is the group algebra of Gn over K. In this paper, we study the faithfulness of the family of representations Φa,b,c in some cases. First, we find necessary and sufficient conditions of the families Φa,0,0,Φ0,b,0 and Φ0,0,c for all n≥2 to be unfaithful, where a,b,c∈K∗. Second, we consider the case n=2 and we find the nature of ker⁡(Φa,b,c) if Φa,b,c is unfaithful. Moreover, we show that there exist some families Φa,b,c that have trivial kernel in the case n=2. Also, we find the shape of the possible elements in ker(Φa,b,c) for all n≥3 when the kernel of Φa,b,c∣SM2 is nontrivial.