co.combinatorics – Circular permutations (bracelets) with similar things (reflections are equivalent) using the enumeration of polia

The circular permutations of N objects of n1 are identical of one type, n2 are identical of another type and so on, so that n1 + n2 + n3 + ….. = N?
There is a similar question but it does not address the case in which the reflections are under the same equivalent class.$$ frac {1} {N} sum_ {d | N} phi (d) p_d ^ {N / d} $$ This is when the reflections are not the same. How does the equation change under this new restriction?

Note: I could not comment on that question because of my low reputation, so I asked this question.