Cadillac CTS-V - Lug nuts on V2 vs V1?

Just to explain, I bought a 2008+ spare tire kit to use the spare on my 1990 Corvette. The tire is the same diameter as my stocker, and the 18" spoked wheel will clear the brakes. My stocker is 20 years old and of dubious safety.

Anyway, so the kit comes with a jack and wrench and whatnot, which I wasn't going to use. But as it happens, the jack is perfect for my 2006 CTS-V once I add a full-size spare.

But the lug handle doesn't work. The hole in it is freaking huge. I thought pretty much all GM cars used a 19mm (or almost 3/4") lug nut.

Do the new V's use some huge ass lugnut? Or is there some kind of adapter you are supposed to use with the thing? Anyone know?

A search shows they are 22mm... So now to see if anyone makes any kind of adapter or something I can toss in. Otherwise I guess I'll need to replace the wrench. Though it is "keyed" to turn the jack.

I thought pretty much all GM cars used a 19mm (or almost 3/4") lug nut. Do the new V's use some huge ass lugnut?

The new V has 14mm lugs vs 12mm on the old V, hence the larger socket size (22mm vs 19mm). It's the same as the '99-up GM trucks, which also requires a 22mm socket.

my v has 19mm lugs!