We investigate how gender stereotypes in authority influence the perceptions and behavior of avatar robots operators and their interlocutors. Gender stereotypes, which typically place men in more authoritative positions than women, are present in not only inter-human but also human-robot interaction. As avatar robots become more integrated into our lives and serve for diverse usages, they may be utilized in positions where they require authority. We study how avatar robot gender and operator gender affect expressions and perception of gender stereotypes in a customer service scenario with 41 pairs of participants. Operators controlled binary gendered avatar robots one at a time, acting as shopkeepers that had to assert authority over customers behaving improperly. The operators perceived their authority to be higher with male avatar robots compared to female ones, regardless of operator gender. We did not detect an effect on customer's perception of the shopkeeper's authority. While less than half of operators and customers perceived authority for reasons related to traditional gender stereotypes, others observed behaviors that did not align with stereotypes. Avatar embodiment may also help operators assert authority safely due to being physically hidden from the customers.