  We study the facility location problems where agents are located on a real
line and divided into groups based on criteria such as ethnicity or age. Our
aim is to design mechanisms to locate a facility to approximately minimize the
costs of groups of agents to the facility fairly while eliciting the agents'
locations truthfully. We first explore various well-motivated group fairness
cost objectives for the problems and show that many natural objectives have an
unbounded approximation ratio. We then consider minimizing the maximum total
group cost and minimizing the average group cost objectives. For these
objectives, we show that existing classical mechanisms (e.g., median) and new
group-based mechanisms provide bounded approximation ratios, where the
group-based mechanisms can achieve better ratios. We also provide lower bounds
for both objectives. To measure fairness between groups and within each group,
we study a new notion of intergroup and intragroup fairness (IIF) . We consider
two IIF objectives and provide mechanisms with tight approximation ratios.
