I’m using the version iOS 16.1.2 and the iMessage doesn’t have an Add Contact at the bottom.
There is no Add Contact on the screen showing the contacts of a group text.
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If there are any non-iPhone users in a group text you cannot add or remove users from the group. This is a limitation of the SMS/MMS network and has nothing to do with your phone or iOS version. And there is nothing Apple can do about it; SMS/MMS is controlled by the mobile carriers.