For \(n\ge 2\) , let \(G_n\) be a group and let \(\rho : B_n\rightarrow G_n\) be a representation of the braid group \(B_n\) . For a field \(\mathbb {K}\) and \(a,b,c\in \mathbb {K}\) , Bardakov, Chbili, and Kozlovskaya extended the representation \(\rho \) to a family of representations \(\Phi _{a,b,c}:SM_n \rightarrow \mathbb {K}[G_n]\) of the singular braid monoid \(SM_n\) , where \(\mathbb {K}[G_n]\) is the group algebra of \(G_n\) over \(\mathbb {K}\) . In this paper, we study the faithfulness of the family of representations \(\Phi _{a,b,c}\) in some cases. First, we find necessary and sufficient conditions of the families \(\Phi _{a,0,0}, \Phi _{0,b,0}\) and \(\Phi _{0,0,c}\) for all \(n\ge 2\) to be unfaithful, where \(a,b,c \in \mathbb {K}^*\) . Second, we consider the case \(n=2\) and we find the nature of \(\ker (\Phi _{a,b,c})\) if \(\Phi _{a,b,c}\) is unfaithful. Moreover, we show that there exist some families \(\Phi _{a,b,c}\) that have trivial kernel in the case \(n=2\) . Also, we find the shape of the possible elements in \(\ker (\Phi _{a,b,c})\) for all \(n\ge 3\) when the kernel of \({\Phi _{a,b,c}|}_{SM_2}\) is nontrivial.
