Well, it's pretty simple. You don't have the condition that they do.
Group riding does take a fair bit of anaerobic condition, it also takes a lot of aerobic base. The riders in the group you are riding with are likely able to ride aerobically at power levels that push you anaerobic.
So you can make some improvement with intervals, but my guess is that it won't be enough. You get aerobic base in the long term, through months or years of riding.
You can keep showing up, but you may still find yourself having issues.
The other point is that late July/August is the worst time to start riding with a new group. Everybody is in top form and riding at their fastest during that time. The best time is March, when even the fast guys are going to be a lot more restrained.