On a fixed point formula of Navarro–Rizo

Let G be a \pi -separable group with a Hall \pi -subgroup H or order n. For x\in H let \lambda (x) be the number of Hall \pi -subgroups of G containing x. We show that \prod _{d\mid n}\prod _{x\in H}\lambda (x^{d})^{\frac{n}{d}\mu (d)}=1, where \mu is the Möbius function. This generalizes fixed point formulas for coprime actions by Brauer, Wielandt and Navarro–Rizo. We further investigate an additive version of this formula.