In this work, we present a study on some properties of the group $VSG_n$, which represents virtual singular braids for $n \geq 2$. We define numerical invariants for the virtual singular braids, obtained through word exponents in $VSG_n$, and describe the kernel of these homomorphisms. We identify all possible homomorphisms from the group $VSG_n$ to the symmetric group $S_n$, considering conjugation. In the particular case where $n = 2$, we present a description and a presentation for the kernel in each case. For all possible homomorphisms, decompositions of $VSG_n$ were obtained as semi-direct products of the kernel of the homomorphism and the symmetric group.