Deubiquitinates histone H2A at 'Lys-13' and 'Lys-15' and regulates DNA damage response. USP51 is recruited to chromatin after DNA damage and regulates the dynamic assembly/disassembly of TP53BP1 and BRCA1 foci. Thiol-dependent hydrolysis of ester, thioester, amide, peptide and isopeptide bonds formed by the C-terminal Gly of ubiquitin (a 76-residue protein attached to proteins as an intracellular targeting signal). Interacts with H2A. Dissociates from chromatin immediately after DNA damage and reassociates with chromatin following DNA repair. Belongs to the peptidase C19 family.