EDIT: @MattWho 's answer made it clear to me I slightly misunderstood the question. His suggestion of managing the entire process using a script is definitely the way to go and would perfectly fit your use-case of someone creating a new instance of an existing process group. If I may add, it sounds like using the nifi registry might benefit you. You could upload your base process group to the registry and use version control with it. Then, when creating new copies, you would instead 'pull' the same process group from the registry instead of creating a copy of process group that exists in your canvas and might unintentionally get changed or deleted.
I do not believe there is a built-in way to directly run an event when a process group is started/stopped, but a reasonable workaround would be to monitor nifi's app-logs and trigger your own event when you receive a log that mentions starting your PG.
If you only want to use tools offered by nifi, you could use the TailFile processor and configure it to run over your app-logs file, then use another processor (such as RouteOnContent) to match the log for starting the PG. From there you can try and do whatever it is that starts the administrative tasks (such as sending an http request with InvokeHTTP processor)
If this answer helped, please mark it as 'solved' and/or apply 'kudos' 🙂