At the very least, the "RN Supervisor" is just that..an RN supervisor. Most often they will be responsible for overseeing the care that the LPNS and CNAs are performing and sometimes other RNs.
Each facility is going to have a different form of job duties assigned to a supervisor. Most places will pay you more than the regular floor or staff nurse, but it might not be that much.
In a lot of places I worked, I could have been the only RN in the building, I assumed responsibiliyt of the entire building. The next higher up in command was the DON.
I can honestly say that when I "supervised" I wasnt labeled as such, I was a Charge Nurse.....but yet I my duties were "supervisor" I never received Extra compensation when I was supervisor.....But everything fell into my lap so to speak.....it was so vague years back...however. I would get a written document stating what your duties are and if a raise is included...it should be....Nowadays I would have the adminstrator and DON sign this and I would make copies.....because in the past even though I made copies of documents myself they had mysteriously disappeared for the Management side. I was in a Non union facility which was tough too, no backup for you. They can lead you around by the nose and just let you go whenever without cause or reason...Very scarey. But this is going back to the mid 90s. So things have changed quite abit since.
Generally the RN Supervisor is just that the Supervisor when the DON or ADON are not in the building. I would ask for the job description from management and what the duties would entail such as are you responsible for taking call, do you still work as a charge nurse at the same time, and the BIG ONE are you responsible for staffing? Good luck.