Texto original - Italiano - fare un barbecue

Does this mean to have a grill party, or to cook something on the barbecue/grill?

CC: apple Ricciodimare Witchy Xini

I can say that it means to cook on a barbecue, since if you're alone cooking on a barbecue we say "fare un barbecue" as well. But notice that it is also used for "have a barbecue party".
Anyway the cooking menaing is ok.