Communications systems to date are primarily designed with the goal of reliable transfer of digital sequences (bits). Next generation (NextG) communication systems are beginning to explore shifting this design paradigm to reliably executing a given task such as in task-oriented communications. In this paper, wireless signal classification is considered as the task for the NextG Radio Access Network (RAN), where edge devices collect wireless signals for spectrum awareness and communicate with the NextG base station (gNodeB) that needs to identify the signal label. Edge devices may not have sufficient processing power and may not be trusted to perform the signal classification task, whereas the transfer of signals to the gNodeB may not be feasible due to stringent delay, rate, and energy restrictions. Task-oriented communications is considered by jointly training the transmitter, receiver and classifier functionalities as an encoder-decoder pair for the edge device and the gNodeB. This approach improves the accuracy compared to the separated case of signal transfer followed by classification. Adversarial machine learning poses a major security threat to the use of deep learning for task-oriented communications. A major performance loss is shown when backdoor (Trojan) and adversarial (evasion) attacks target the training and test processes of task-oriented communications.