Equivariant Mobius inversion

Sami Assaf and I prove this in section 5 of our paper Specht modules decompose as alternating sums of restrictions of Schur modules. It is surprising that we couldn't find a reference!

This is more of a long comment. I am not sure I understand your construction, but the sort of alternating sum you take has a preimage in the Burnside ring, and is often called the "Lefschetz invariant" by finite group theorists. One of the Representation and Cohomology books by Benson has a section about this. Also see this paper by Thévenaz.