Different from data-oriented communication systems that primarily focus on how to accurately transmit every bit of data, task-oriented semantic communication systems only transmit the specific semantic information required by downstream tasks, strive to minimize the communication overhead and maintain competitive tasks execution performance in the presence of channel noise. However, it is worth noting that in many scenarios, the transmitted semantic information needs to be dynamically modified according to the users' preferences in a conversational and interactive way, which few existing works take into consideration. In this paper, we propose a novel cross-modal editable semantic communication system, named Editable-DeepSC, to tackle this challenge. By utilizing inversion methods based on StyleGAN priors, Editable-DeepSC takes cross-modal text-image pairs as the inputs and transmits the edited information of images based on textual instructions. Extensive numerical results demonstrate that our proposed Editable-DeepSC can achieve remarkable editing effects and transmission efficiency under the perturbations of channel noise, outperforming existing data-oriented communication methods.