Hi GaryMZT.
Welcome to the Microsoft Community.
I read your question carefully and your Phone Link looks basically normal. Based on your description, it looks like your messages will be sent normally and also show up on your phone, the problem looks like only this group icon. You can try sending your messages using the SMS program that came with your phone instead of a third party SMS program. Sometimes third-party SMS programs have problems.
If this issue affects your normal usage. Unfortunately, there aren't a lot of settings in the program that can be tweaked by the user. I would love to be able to help, but I am not a Microsoft developer. You can give feedback on your problem at the website below, please be as detailed as possible in describing the problem you are experiencing. Microsoft developers see every user's feedback and your feedback is very important to us.
Send feedback to Microsoft with the Feedback Hub app - Microsoft Support
Have a Happy New Year!
Best Regards,
Raylan-MSFT | Microsoft Community Support Specialist