I just did this fix two days ago on my 5-speed. The old switch had died when the plastic top section broke off. There was nothing to hold the spring in place, which keeps pressure on the switch plunger. This one screwed directly into the extension, with only an extra nut to tighten it in place. No extra bracket.
You can take the switch out from the top, but putting a new one back in is tough from up there. I had to come from below to get the thread started, then tightened it down from inside the car. Once it is near where you think it belongs, plug in the wiring loom, turn on the ignition switch, put the car in reverse and screw in the switch until the backup lights come on. Drop it out of reverse and check that they go off. If not, adjust the position until they do, then tighten down the nut that holds the switch in place.
Put it all back together and you are done!
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1980 Pageant Blue DHC with removable hardtop.