You can look at the open source software called “Open Source Ticket Request System” which would give you what you should want.
Bugzilla is another popular ticketing system.
Whilst not free, I would recommend FocalScope. It is great for us because it offers email routing & group management tools which let us define responsibility groups and get the right emails/tickets to them for resolution. It also offers us internal chat, live chat from our website, tools for integrating SLAs and an easy to learn interface.