In this paper, we explore an invertible operator \( B \) on \( l^2(\mathbb {Z}_N) \) that maps a regular Gabor frame for \( l^2(\mathbb {Z}_N) \) to another regular Gabor frame for \( l^2(\mathbb {Z}_N) \) while preserving the same index set, even though \( B \) does not commute with the involved translation and modulation operators. We establish both the existence and the matrix representation of such an invertible operator on \( l^2(\mathbb {Z}_N) \) .
