Are you referring to iMessage, or regular group text? Text messages, and group messages, and MMS are carrier responsibiliy. Once you start a group message, some carrier treat those as MMS and you are required to have an MMS plan. Even a group iMessage is turned into an MMS. Check with your carrier if you are having issues with sending.
Why you insist in saying MMS ? MMS stands for "Multimedia Messaging Service" and has absolutely nothing to do with the issue.
By the way, having the same issue that keeps creeping in, I turned it off so only SMS - Short Message Service is enabled.
Of course, SMS also has nothing to do with group messaging either
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Actually it does, dependent on the carrier. Many carriers treat group messages, even those without any media attached as MMS. That also explains why with many carriers the iPhone will not allow you to turn on Group Messaging unless you turn on MMS. The button/switch is missing in Settings>Messages unless you have MMS turned on. It doesn't matter anyway, but I'm glad you have a working pos.