Hey there Sally504,
It looks like you joined the Apple Support Communities from a great resource; Use Messages on your iPhone or iPad - Apple Support. However, for the information you're looking to get, we have a few other links that may help answer your questions more directly.
Check out this link to learn how to add or remove contacts from a group text message on your iPhone: Add and remove people in group text messages on your iPhone or iPad - Apple Support.
How to add someone to a group text message
1. Tap the group message that you want to add someone to.
2. Tap the group icons at the top of the thread.
3. Tap the gray arrow icon to the right of the contacts, then tap Add Contact
.
4. Type the contact that you want to add, then tap Done.
You can then look over the information on this page to learn how to change the name of your group message: Name a group text message on your iPhone, iPad, or iPod touch - Apple Support. As you can see there, you can even add a photo so the group is easier to find.
We hope this information helps you out.
Best wishes!