Can argparse do mutual exclusion of groups instead of just individual options?

Question

I would like to build an ArgumentParser object with two groups of arguments that are mutually exclusive to one another, like so

usage: obtain_pizza [-h] --topping TOPPING ( --pizzeria PIZZERIA --budget BUDGET | --check-fridge-only )

I attempted to do this by just adding a group inside a mutual exclusion group, like so:

parser = argparse.ArgumentParser()
parser.add_argument('--topping', nargs=1, required=True)

mutually_exclusive_group = parser.add_mutually_exclusive_group(required = True)

purchase_group = mutually_exclusive_group.add_argument_group('Purchase options')
purchase_group.add_argument('--pizzeria', nargs=1)
purchase_group.add_argument('--budget', nargs=1)

mutually_exclusive_group.add_argument('--check_fridge_only', action='store_true')

However this results in the program apparently behaving as though purchase_group were at top level:

usage: obtain_pizza.py [-h] --topping TOPPING 
       [--pizzeria PIZZERIA] [--budget BUDGET]
       --check_fridge_only

(line broken up for readability)

Note: I see a similar question at Set up mutually exclusive sets in argparse but it's nine years old and doesn't have an answer other than using third-party packages. My question is about doing this using argparse specifically.