Back in my film days, I, too, had a 2X 'extender' that converted my 100-200 to 200-400. I think it was a Vivitar, as I recall, and I bought it at the same time as I bought the 100-200, also a Vivitar. The extender then, as now, mounts between the camera and the lens and had the correct mounting hardware to mate with both. 3rd party manufacturers make their lenses in a variety of mounts to broaden their market. Other than a couple of "let's see what it'll do" type of tests, I didn't use the 2x much. I did, however, get some good shots of small snow slides on Mt Ranier near Seattle on a 95 degree day using the 100-200 with the 2X. I don't think I ever used it again after that. The 100-200 didn't see much use either.
These days, most extenders are either 1.4x or 2.0x and still mount between the lens and the camera, providing the proper mount and electrical connections as well. The better ones cost several hundred dollars each. There are also a number of never-heard-of-them manufacturers that market cheap, screw-on 'telephoto extenders' or whatever they choose to call them that are complete junk. Typically, they are less than $40. These screw into the filter threads and don't work very well, producing an image that is far from being sharp. I tried one, thinking, what the heck?' The real lens couldn't focus in AF mode, so I focused manually. The results were blurry, and while removing the 'telephoto extender', the whole thing fell apart! So, don't waste your money on that trash.