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https://www.doc.govt.nz/parks-and-recreation/things-to-do/hunting/permits-and-licences/permit-conditions/

Under excluded list
4.0 The following activities are strictly prohibited:
...

And even the 22 hornet with the new powders throwing a 50 gr bullet at >2700 fps, or 55 gr at 2650 fps.
Is definitely a no no as are all the 17 and 20 cal cartridges.
Z