This cannot be done in a private session. I won 5 Flag Rush private games and never got it. However, as soon as I went to a public online game I was able to get it with no problem.
Just start an online free roam session while driving one of the Extreme Off Road vehicles, select one of the playground games icons and create a blue print for an Extreme Offroad flag rush game. Start the game and win it by capturing and/or stealing more flags than anyone else and getting them to their respective goals within the time limit to satisfy this challenge.
Of course this presumes someone will accept your playground game challenge when you start it. If nobody does, the challenge request times out and you'll have to keep trying until someone in the session does accept. If this is a problem you might try having a friend join you online.
If you don't know how to find an Extreme Off Road vehicle from all the others, select garage cars from the Auto show. Once they are displayed, press Y to filter, and then scroll down the options until you see Extreme Off Road. Click on the radio button and back out with the B button. Only those vehicles that match your category will show. If none show, use the same filter method to find one from the cars available to buy.