Hello fa6664,
We understand that the Apple Messages app isn't pulling up the name of a group when you create a new message. We'll do our best to assist with this.
Do you still have the group message on your iPhone? If so, is the group message titled or named? If not, we'd recommend naming the group message, then try to search for the name of the group message in your new conversation: Name a group text message on your iPhone, iPad, or iPod touch - Apple Support
How to name a group text message
You can name a group iMessage as long as everyone is using an Apple device, such as an iPhone, iPad or iPod touch. You can’t name SMS/MMS group messages or iMessage conversations with just one person.
1. Open the group iMessage
Tap the group icons at the top of the message thread. If you have iOS 14 or earlier, tap the icons at the top of the thread, then tap the info button.

2. Tap Change Name and Photo
The Change Name and Photo option will be in blue text under the group icons and names of the people on the thread.
Remember, you can name group iMessages, but not SMS/MMS group messages. To name a group text message, everyone needs to be using an iPhone, iPad or iPod touch.

3. Add a name and photo for the group
After you've entered the name for the group, tap the Camera button to take a photo. Or select a photo you already have. You can also choose an emoji or Memoji for your group photo.
When you’ve finished, tap Done.

In case the issue continues to occur, does it happen with specific group messages or names? Are you able to search in the Apple Messages app without issues to send other messages or find content?
Let us know.
Cheers!