Maybe you could generate all 'pairs' (which scales nicely to N) then filter them:

use strict;
use warnings;

my $str = 'aaAabBAAa';
my $N = 2;

print join "\n", grep {/^(.)\1+$/i}
    map {substr $str, $_, $N} 0 .. length ($str) - $N;
[download]

Prints:

aa
aA
Aa
bB
AA
Aa
[download]

Update: Changed to match N adjacent characters.