Any app will only input its own info, so its own list of contacts. This would not effect the info for the phone contacts, which is different.
If you set up the groups via your Gmail account, and not created the groups on the phone itself, then the groups would stay in tact and you can save them to any device that is connected to that account.
You can save the contacts from the phone to the Gmail account, or save them to a CSV file and put it on a computer. Then upload that file to the Gmail account the phone uses. Be sure you clear the contacts completely off the phone after you have made the copy, but before putting them in the Gmail account. If you need to put them in the Gmail account first (rather than a CSV file, but a direct...