The TC-16a has AF in the same way a normal screw-drive AF lens works, it won't focus unless the camera has a built-in AF motor.
On the other hand, the original TC-16 (no "a") for the F3-AF had the AF motor built into the TC. It's a rather bulky unit, even though the AF motor is quite small since it just needs to drive small elements in the TC:
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As Ilkka suggested, it could be possible that Nikon produces more than one F-mount adapter:
(V1) The simplest would be an all-electronic "E" adapter. It would provide AF only with AF-S, AF-P and AF-I lenses (no screw-drive AF) and automatic aperture only on lenses with the electronic "E" diaphragm. Since it has no mechanical linkage, the adapter could also support stop-down metering with other lenses with a manual aperture ring, so could be usable with pre-AI, AI and early AF lenses. It would not be compatible with G lenses. My feeling is that Nikon would consider this adapter too limited so would not go for this option.
(V2) More likely is an adapter as above which also has a built-in motor so that it can stop down lenses with the mechanical aperture linkage - which is the vast majority of F-mount lenses. It would AF with AF-S, AF-P and AF-I lenses (no screw-drive AF) and would support metering and automatic aperture with CPU lenses. However it would not support metering with other lenses. If the mechanical linkage could be disconnected it stop-down metering could be a possibility.
In theory it could also support metering and automatic aperture with AI-S lenses, provided lens CPU data is entered in the camera, the aperture is set via the camera (not lens), and the lens aperture ring is set to minimum aperture, but it would need to detect the AI-S indicator "scoop" on the lens. It would not work with AI and older lenses due to the non-linear aperture action. Given that Nikon has not made a camera which distinguishes AI from AI-S since the late 1980s, this seems unlikely.
(V3) If V2 also has an AI linkage, it would also offer metering with AI lenses. It would not be compatible with pre-AI lenses unless the AI tab could be flipped up.
(v4) The most comprehensive, complex, and expensive adaptor would also have a built-in AF motor like the old TC-16, to support screw-drive AF. This would provide backward compatibility with almost every F-mount lens made...