Interval-value Fermatean hesitant fuzzy sets (IVFHFSs) are new data model for dealing with complex, uncertain information. However, the researches on IVFHFSs score functions (SFs) have low discrimination rate and existing multi-attribute group decision-making under IVFHFSs are scarce and have no ability to classify. Therefore, this paper establishes a new three-way multi-attribute group decision-making (3W-MAGDM) based on IVFHFSs. First, novel IVFHFSs SFs are proposed. Next, the objective conditional probabilities calculation method is derived by using the probabilistic dominance relation and the relative loss functions calculation method in three-way decision is developed. The subjectivity of 3W-MAGDM under IVFHFSs is greatly reduced. Finally, a new 3W-MAGDM framework based on IVFHFSs is constructed. The new approach has a high discrimination rate in SFs and it not only has a ranking function but also has a categorization function.