Catalysis of the reaction: 1,3-alpha-D-mannosyl-1,2-alpha-D-mannosyl-1,2-alpha-D-mannosyl-D-mannose + UDP-N-acetyl-D-glucosamine = 1,3-alpha-D-mannosyl-1,2-(N-acetyl-alpha-D-glucosaminyl-alpha-D-mannosyl)-1,2-alpha-D-mannosyl-D-mannose + UDP.
UDP-N-acetyl-D-glucosamine:mannotetraose alpha-N-acetyl-D-glucosaminyltransferase activity