Glycosyltransferase that mediates core 2 O-glycan branching, an important step in mucin-type biosynthesis. Does not have core 4 O-glycan or I-branching enzyme activity. O(3)-[beta-D-galactosyl-(1->3)-N-acetyl-alpha-D-galactosaminyl]-L-seryl-[protein] + UDP-N-acetyl-alpha-D-glucosamine = 3-O-{beta-D-galactosyl-(1->3)-[N-acetyl-beta-D-glucosaminyl-(1->6)]-N-acetyl-alpha-D-galactosaminyl}-L-seryl-[protein] + H(+) + UDP O(3)-[beta-D-galactosyl-(1->3)-N-acetyl-alpha-D-galactosaminyl]-L-threonyl-[protein] + UDP-N-acetyl-alpha-D-glucosamine = 3-O-{beta-D-galactosyl-(1->3)-[N-acetyl-beta-D-glucosaminyl-(1->6)]-N-acetyl-alpha-D-galactosaminyl}-L-threonyl-[protein] + H(+) + UDP Protein modification; protein glycosylation. Predominantly expressed in thymus. Weakly expressed in pancreas, peripheral blood leukocytes, placenta, small intestine and stomach. Barely detectable in liver, spleen, lung and lymph node. Belongs to the glycosyltransferase 14 family. Core 2 beta-1,6-N-acetylglucosaminyltransferase 3