Missed Chats assigned to task in CRM

Missed Chats assigned to task in CRM

Anyone we miss at chat with, there is a task scheduled in CRM. That would be great, but the task is just a note of the original question with nothing attaching it to lead or a contact. Almost like it is an empty task with a date. The only way to find the lead or contact in CRM is to search through previous chats, find the one that has the comment in the "note" of the task and see if a name, email, or phone number is provided. If that information was provided, we can then search CRM for that information and appropriately task that lead.

Is there a way to have a missed chat be assigned as a task to an individual that attaches the missed chat transcript (we are using a 30 second proactive chat trigger on the site) which would have their name and email and that designated person would be notified by email of this task?