Alright look. I dont have dedicated ventilation system so Im going to be pump a 4inch & 6inch exhaust out a window.
I live in a apartment.. Can't be cutting new holes or going up or down. In between people lmFAO!
If you look up info on portable A/C's, they have a piece that fits into the window, and the window shuts on it, the same as that fan. Then it would just look like you had a portable A/C. They usually have 1 or 2 openings approx. 4" in diameter. You would just need to make sure it is light-proof, and has some sort of carbon filter for smell.
I suggest a piece of plywwood with holes cut out to fit the ducting. This way if u only want ur window open 10" u cut the ply 10". You can also do this with cardboard, but cardboard will allow cold air in. You can also use the cardboard for a trial run, that way u can experiment with wat u want, and when u make ur decision, the final product can be plywood. They make hole saws that will cut these dimentions and attach to a cordless drill. Good luck! They also sell a foam insulation that is between an inch or two thick that is easy to tack on the ply. You can paint the outside of the ply and use dryer vents in the holes of the ply. This way u will be able to attach the ducting. I dont see this costing more than $30. I would almost use that double window fan as a front. You could glue ar attach the plywood with the dryervents already attached, and booya. No one knows unless they walk by. For smell, u can take cheese cloth put coffee grinds in between the cheese cloth, take the cheese cloth and put in the ducting, then attach the ducting to ur ply or watever u are using for the window. I hope u understand this, I do not type well