The inner loop can be replaced with a hash to change the algorithm from O(N²) to O(N), a major improvement.

my @groups;
my %in_num_groups;
while (@pile) {
    my $item = shift @pile;
    push @{$groups[$in_num_groups{$item}++]}, $item;
}
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I also fixed the needless use of a reference to the array of groups and the inability to store false values in @pile.

Update: For fun, did you know @pile could also be initialized using

@pile = shuffle( (@card) x $n );
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