What game did I see in Verona, Italy?

During October 2016 I was in Verona, Italy. I visited the Torre dei Lamberti. Within the square there was a game, perhaps a tournament as there was two matches going on at once.

What is this game?

It was hard to tell exactly what was going on, but I remember these details.

• It appeared to be 4 a side.
• Within a single round there was 4 'fielders' versus 1 'batter', with rotations between rounds.
• The batter has a wooden bat, and starts with a wooden peg. They are stood next to a large round stone. The batter tosses wooden peg into the air and attempts to hit it as far as they can with the bat.
• The fielders take up 'fielding' positions to either catch or block the peg. The fielders also appear to try and distract the batter.
• In some instances the batter then gets to hit the peg again. In which case they have to first make the peg jump off the ground but hitting one end of it, and then attempt to hit it whilst it's in the air.
• In some instances one of the fielders will attempt to throw the peg back at the large stone. The batter will attempt the defend the stone with their bat.
• The game playing area was surrounded by talls nets.
• Two games took up the whole square with each game having roughly half of the square. The pitch was rectangluar.
• The players were dressed in normal clothing, but all had a matching scarf.
• The peg was a cylinder with tappered ends so it never sat flat on the ground.