In this paper, we present a study aimed at understanding whether the embodiment and humanlikeness of an artificial agent can affect people's spontaneous and instructed mimicry of its facial expressions. The study followed a mixed experimental design and revolved around an emotion recognition task. Participants were randomly assigned to one level of humanlikeness (between-subject variable: humanlike, characterlike, or morph facial texture of the artificial agents) and observed the facial expressions displayed by a human (control) and three artificial agents differing in embodiment (within-subject variable: video-recorded robot, physical robot, and virtual agent). To study both spontaneous and instructed facial mimicry, we divided the experimental sessions into two phases. In the first phase, we asked participants to observe and recognize the emotions displayed by the agents. In the second phase, we asked them to look at the agents' facial expressions, replicate their dynamics as closely as possible, and then identify the observed emotions. In both cases, we assessed participants' facial expressions with an automated Action Unit (AU) intensity detector. Contrary to our hypotheses, our results disclose that the agent that was perceived as the least uncanny, and most anthropomorphic, likable, and co-present, was the one spontaneously mimicked the least. Moreover, they show that instructed facial mimicry negatively predicts spontaneous facial mimicry. Further exploratory analyses revealed that spontaneous facial mimicry appeared when participants were less certain of the emotion they recognized. Hence, we postulate that an emotion recognition goal can flip the social value of facial mimicry as it transforms a likable artificial agent into a distractor.