I am the GIS manager for a city. I am under the Information Systems Dept. My thought is to evaluate and take note of how the current maintenance programs are conducted for all the Information Systems in your city. Is IT dept maintaining a server room with several servers? Is there already a staff of people that maintain these servers? Any dedicated GIS Staff in the IT Dept and/or in any other dept?
If you are looking at an enterprise approach, first determine if it will be maintained locally or in the cloud. Research this?
Second, since this will implemented across many departments, figure out if you will use a Centralized environment or a decentralized environment or maybe both. The way I am using these terms here is if the individual departments will be able to maintain their authoritative data or not. I have both environments. For example, at the city that I work in the Planning dept maintains there address point layer. I call this decentralized environment because they make edits to the Address point layer. If they did not have the man power, resource, or skill level to maintain this GIS Layer then the GIS Division under the IS Department would resume responsibility.
So my suggestion of "should the GIS be put managed by another department?" I believe it should be under the IT dept. Even more so if the City goes to an Enterprise Level.
Of course there is more thought to put into this but I believe this is a good starting point. I hope this helps.