While machines learn from existing corpora, humans have the unique capability to establish and accept new language systems. This makes human form unique language systems within social groups. Aligning with this, we focus on a gap remaining in addressing translation challenges within social groups, where in-group members utilize unique terminologies. We propose KpopMT dataset, which aims to fill this gap by enabling precise terminology translation, choosing Kpop fandom as an initiative for social groups given its global popularity. Expert translators provide 1k English translations for Korean posts and comments, each annotated with specific terminology within social groups' language systems. We evaluate existing translation systems including GPT models on KpopMT to identify their failure cases. Results show overall low scores, underscoring the challenges of reflecting group-specific terminologies and styles in translation. We make KpopMT publicly available.