The automatic identification of hate speech constitutes an important task, playing a relevant role towards inclusivity. In these terms, the shared task on Climate Activism Stance and Hate Event Detection at CASE 2024 proposes the analysis of Twitter messages related to climate change activism for three subtasks. Subtasks A and C aim at detecting hate speech and establishing the stance of the tweet, respectively, while subtask B seeks to determine the target of the hate speech. In this paper, we describe our approach to the given subtasks. Our systems leverage transformer-based multi-task learning. Additionally, since the dataset contains a low number of tweets, we have studied the effect of adding external data to increase the learning of the model. With our approach we achieve the fourth position on subtask C on the final leaderboard, with minimal difference from the first position, showcasing the strength of multi-task learning.