How do I remove the center cap and re-install on a bundt wheel?

Remove the wheel and use something to push it out from the back.

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Milan Brown 1979 240D, rebuilt OM617.952 turbo diesel, rebuilt 722.315 transmission (my only daily driver)

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Can I hit a center punch with a hammer to knock it out? I don't want to damage it.

I use the wooden handle of a regular hammer to just tap it out from behind after the wheel is off, comes out real easy.

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1980 300TD, SOLD
1984 300TD, 275K
1999 C230 K Black & Tan
2013 C250 Black
1974 CJ5 Red
You might faint from the fight, but you're gonna find it. Every challenge could have paradise behind it. -John Popper-

There is at least 2 type. One type has metal fingers that spread out into a recess in the wheel and the other type has plastic fingers with a near circular C shaped spring to spring load the plastic fingers.

What was mentioned is the safest method but if you are extremely carful you can get a sturdy putty knife between the wheel and the center part and pray it out enough to get a screwdriver under it. And then slide the screwdriver around it prying it out a bit till there is enough of it to grab and pull off. You don't need to remove the wheel for that.

The ones with the plastic fingers are slightly harder to get out as the plastic grabs.

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84 300D, 82 Volvo 244Gl Diesel