Weights for $\pi$-partial characters of $\pi$-separable groups

The aim of this paper is to confirm an inequality predicted by Isaacs and Navarro in 1995, which asserts that for any ${\pi }^{\mathrm{\prime }}$-subgroup $Q$ of a $\pi$-separable group $G$, the number of ${\pi }^{\mathrm{\prime }}$-weights of $G$ with $Q$ as the first component always exceeds that of irreducible $\pi$-partial characters of $G$ with $Q$ as their vertex. We also give some sufficient condition to guarantee that these two numbers are equal, and thereby strengthen their main theorem on the $\pi$-version of the Alperin weight conjecture.