When you say "group of users" do you mean in the Admin Console... you created a permission group?
I don't think Alerts works like that... it only works with emails. (Despite appearances of having your own user added as an object, I don't think this interface was ever updated to support typing in a user or a user group with auto-suggest to allow you to select people that way... Honestly, that would be a great feature...💡)
Here are some options for right now... if your IT department can create mailing list emails, then you can send alerts to that type of email.
Or, if you use Slack or MS Teams, you can create channels... then you can actually get emails for those channels, and send alerts right into those tools... so your users can actually join those channels and get the alerts there, instead of your inbox.