Catalysis of the reaction: N-acetyl-beta-D-glucosaminyl-1,3-N-acetyl-D-galactosaminyl-R + UDP-N-acetyl-D-glucosamine = N-acetyl-beta-D-glucosaminyl-1,6-(N-acetyl-beta-D-glucosaminyl-1,3)-N-acetyl-D-galactosaminyl-R + UDP.
UDP-N-acetyl-D-glucosamine:O-oligosaccharide-glycoprotein (N-acetyl-D-glucosamine to N-acetyl-D-galactosamine of N-acetyl-beta-D-glucosaminyl-1,3-N-acetyl-D-galactosaminyl-R) beta-1,6-N-acetyl-D-glucosaminyltransferase activity