Let $\mathfrak{D}$ be a residually finite Dedekind domain and let $\mathfrak{n}$ be a nonzero ideal of $\mathfrak{D}$. We consider counting problems for the ideal chains in $\mathfrak{D}/\mathfrak{n}$. By using the Cauchy–Frobenius–Burnside lemma, we also obtain some further extensions of Menon's identity.