I was deployed to this particular AO for approximately ten months, back in 2016-2017.
The threat profile changed dramatically during that time period; in June 2016, ISIL was less than 50km from Baghdad, Fallujah was still raging, protests threatened to topple the government and repeated SVBIEDs in the capital (mostly Shi'ite areas) were a daily occurrence. By the time I left the following spring in 2017, the Euphrates River Valley was largely clear, Mosul was half-done and threats from separate actors, including drones, were a concern. I'd be interested to see what, if anything, has changed in the area, but ISIL would be largely seen as a minor point.
This is only supposition and conjecture, but if you want a view of what we would LIKELY be doing, review TF CARABINIERI at Camp Dublin, or the CJTF-OIR Building Partner Capacity site information, available in public forums.
Headquarter personnel would be operating, as with any higher formation, from all trades and be occupying all manner of staff jobs; it will be interesting, from a Canadian perspective, to observe how NATO relationships occur within the spectrum of CJTF-OIR, JTF-I (Canadian) and Iraqi Security Forces. It may be...complex.