Back out the screws holding the fan, filter and electrical box inside the vent assembly by turning the screws counterclockwise, and then remove those parts from the assembly. Hold the vent assembly in its slot above the range and mark the location of the openings in the back of the assembly on the wall.
Remove the vent assembly and use a drill to make holes at the corners of the duct outline you drew, drilling all the way through the wall and to the outside of the house. Drill a hole where you marked the vent’s electrical connection.
Draw straight lines between the holes you drilled, on the exterior of the house. Cut along the lines with a jigsaw, making a cutout for the exhaust duct. Insert the fan’s duct into the hole you cut until the cap presses firmly against the outside of the house. Drive screws through the mounting holes on the perimeter of the cap, securing it to the outside of the house.
Turn off the power to the kitchen by flipping the breaker switch on the house’s electrical panel. Pull a wire from a nearby junction box to the hole you drilled for the vent assembly’s electrical wiring. Use a wire stripper to remove about 2 inches of the insulation from the wires.
Feed the stripped wires through the electrical opening in the vent assembly, tightening in the screw in the opening to hold the wires in place. Replace the vent hood over the range top, lining up the exhaust duct with the opening in the back of the hood. Drive screws through the mounting holes in the back or sides of the hood, securing it to the wall or cabinets.
Twist the same colored wires from the wiring in the wall with the wiring inside the vent hood, twisting on wire nuts to secure the connections. Replace the parts you removed from the vent hood originally, tightening the screws to hold the parts in place.