On twisted cohomology, equivariant cohomology and (stable) twisted/equivariant Cohomotopy:
James Cruickshank, Twisted Cobordism and its Relationship to Equivariant Homotopy Theory, 1999 (pdf, pdf)
(the equivariant Pontrjagin theorem for free G-manifolds)
James Cruickshank, Twisted homotopy theory and the geometric equivariant 1-stem, Topology and its Applications Volume 129, Issue 3, 1 April 2003, Pages 251-271 (doi:10.1016/S0166-8641(02)00183-9)
(Lemma 5.2 is the twisted Pontrjagin theorem)
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