Molecular cavity optomechanics (COM), characterized by remarkably efficient optomechanical coupling enabled by a highly localized light field and ultrasmall effective mode volume, holds significant promise for advancing applications in quantum science and technology. Here, we study optomechanically induced transparency and the associated group delay in a hybrid molecular COM system. We find that even with an extremely low optical quality factor an obvious transparency window can appear, which is otherwise unattainable in a conventional COM system. Furthermore, by varying the ports of the probe light, the optomechanically induced transparency or absorption can be achieved, along with corresponding slowing or advancing of optical signals. These results indicate that our scheme provides an alternative method for adjusting the storage and retrieval of optical signals in such a molecular COM device.