Some remarks on Kida's formula when $\mu \mathrm{\ne }0$

The Kida’s formula in classical Iwasawa theory relates the Iwasawa $\lambda$-invariants of $p$-extensions of number fields. Analogue of this formula was subsequently established for the Iwasawa $\lambda$-invariants of Selmer groups under an appropriate $\mu =0$ assumption. In this paper, we give a conceptual (but conjectural) explanation that such a formula should also hold when $\mu \mathrm{\ne }0$. The conjectural component comes from the so-called ${\mathfrak{M}}_H(G)$-conjecture in noncommutative Iwasawa theory.