Let R∈C^(n×n) be a {k}-involutory matrix (that is, R^k=I_n) for some integer k≥2, and let s be a nonnegative integer. A matrix A∈C^(n×n) is called an {R,s+1,k}-potent matrix if A satisfies R A = A^(s+1) R. In this paper, a matrix group corresponding to a fixed {R,s+1,k}-potent matrix is explicitly constructed, and properties of this group are derived and investigated. This group is then reconciled with the classical matrix group G_A that is associated with a generalized group invertible matrix A. Let R∈Cn×n be a {k}-involutory matrix (that is, Rk=In) for some integer k≥2, and let s be a nonnegative integer. A matrix A∈Cn×n is called an {R,s+1,k}-potent matrix if A satisfies RA=As+1R. In this paper, a matrix group corresponding to a fixed {R,s+1,k}-potent matrix is explicitly constructed, and properties of this group are derived and investigated. This group is then reconciled with the classical matrix group GA that is associated with a generalized group invertible matrix A.