Catalysis of the reaction: UDP-N-acetyl-D-glucosamine + beta-D-galactosyl-(1->3)-N-acetyl-D-galactosaminyl-R = UDP + beta-D-galactosyl-(1->3)--N-acetyl-D-galactosaminyl-R. This reaction is the addition of N-acetyl-alpha-D-glucosamine to the core 1 structure of O-glycans forming core 2.
beta6-N-acetylglucosaminyltransferase activity