Use a radon detection system to measure the amount of radon in your basement or crawl space. Anything over 4 PC (Pico Curies per liter pCi/L) is considered by the EPA as dangerous and requiring radon mitigation.
Inspect the outside location of the target downspout and find the nearest point for the PVC exhaust line. Measure the distance from a known point that can be referenced in the basement or crawl space. Record that distance.
Inspect the basement or crawl space in your home to determine if it is vented with outside air. If it is not vented, determine the best vent location -- on an opposite wall from the target downspout if possible -- and install the vent in the 2x12 rim joist. Trace the outline of the vent, drill a hole in the joist and cut along the outline with a keyhole saw. Install the vent with 2-inch wood screws.
Install the inline radon exhaust fan by building a 2-inch-by-4-inch pine board support platform for it to sit on. Construct the support platform so there is open space in the center that the fan's underside vent coupler (air inflow vent) fits snugly into. Cover the import vent coupler with plastic window screening, seal it with the band clamp and install the fan into the platform. Attach the platform as closely as possible to a rafter or sill near the target downspout. Finalize the installation by wrapping HVAC duct support strapping around the fan and nailing it to the nearest rafter.
Use the PVC coupler, PVC pipe and PVC elbows to construct a pipeline to the nearest exit point in the rim joist next to the downspout. Cut the PVC pieces with a keyhole saw and test fit them to determine the success of your cuts. Use the PVC cleaner to prepare all the fittings and connect them with PVC glue. Attach the piece nearest the fan to the fan using the manufacturer-supplied PVC coupler or your own.
Mount a 110-volt switch-control junction box on a wooden support member near the fan and tie it into the home's power panel. Wire the fan to the junction box.
Drill a 4-inch hole in the rim joist for the PVC pipe with a power drill and hole-saw bit. Run the exhaust line out the hole and seal both sides with rubber cement. Use PVC elbows and pipe to get it to reach the downspout.
Drill a hole in the side of the downspout above any terminal angles or connectors. Use a keyhole saw to cut a hole big enough to accommodate connection with the exhaust pipe. Frame a connection from the PVC pipe to the downspout using 5-inch HVAC flex duct. This part varies as downspouts differ in size and shape. Use duct tape if necessary.
Test the system by powering up the fan and allowing it to run for several hours. Hold your hand under the fan and you should feel good airflow going into it. Then hold your hand above the downspout/gutter connection and you should feel good airflow there.