Reconfigurable Intelligent Surfaces (RIS) hold the promise of improving significantly coverage, as well as spectral and energy efficiency in wireless communication systems. Techniques based on RIS form a key technology for 6G systems. An important issue in RIS technology is Channel State Information (CSI), which is much more difficult to acquire in such systems. This work introduces a Differential Space-Time Modulation (DSTM) scheme integrated with Differential Reflecting Modulation (DRM) to bypass the requirement for CSI in such systems, while providing error rate gains. The DSTM scheme is based on unitary group codes. We first consider uncoded DRM for RIS to serve as a reference point. Next we provide an overview of DSTM and outline the procedures for its integration with DRM. Furthermore, we explore the extension of both the original DRM and the coded DRM-DSTM scheme to a larger number of RIS reflecting patterns $K$, and provide tables of codes for $K= 2, 3, 4$. Encoding and decoding complexities are studied as well. Extensives simulation results over quasi-static Rayleigh fading channels confirm the effectiveness of the DRM-DSTM coded system, illustrating its advantages over uncoded DRM with proper system parameters.