Monolithic partial characters

Let $\pi$ be a set of prime numbers, $G$ be a $\pi$-separable group, and $H$ be a Hall $\pi'$-subgroup of $G$. We prove in this note that $H$ is normal in $G$ and $G/H$ is nilpotent if and only if $\varphi(1)^{2}$ divides $|G: \ker\varphi|$ for all monolithic partial characters $\varphi\in {\rm I}_{\pi}(G)$, where ${\rm I}_{\pi}(G)$ is the set of irreducible partial characters of $G$.