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There is a broad consensus on the importance of deep learning models in tasks involving complex data. Often, an adequate understanding of these models is required when focusing on the transparency of decisions in human-critical applications. Besides other explainability techniques, trustworthiness can be achieved by using counterfactuals, like the way a human becomes familiar with an unknown process: by understanding the hypothetical circumstances under which the output changes. In this work we argue that automated counterfactual generation should regard several aspects of the produced adversarial instances, not only their adversarial capability. To this end, we present a novel framework for the generation of counterfactual examples which formulates its goal as a multi-objective optimization problem balancing three different objectives: 1) plausibility, i.e., the likeliness of the counterfactual of being possible as per the distribution of the input data; 2) intensity of the changes to the original input; and 3) adversarial power, namely, the variability of the model's output induced by the counterfactual. The framework departs from a target model to be audited and uses a Generative Adversarial Network to model the distribution of input data, together with a multi-objective solver for the discovery of counterfactuals balancing among these objectives. The utility of the framework is showcased over six classification tasks comprising image and three-dimensional data. The experiments verify that the framework unveils counterfactuals that comply with intuition, increasing the trustworthiness of the user, and leading to further insights, such as the detection of bias and data misrepresentation.