Given a selfmap $f:X\to X$ on a compact connected polyhedron $X$, H. Schirmer gave necessary and sufficient conditions for a nonempty closed subset $A$ to be the fixed point set of a map in the homotopy class of $f$. R. Brown and C. Soderlund extended Schirmer's result to the category of fiber bundles and fiber-preserving maps. The objective of this paper is to prove an equivariant analogue of Brown-Soderlund theorem result in the category of $G$-spaces and $G$-maps where $G$ is a finite group.
"Fixed point sets of equivariant fiber-preserving maps." Bull. Belg. Math. Soc. Simon Stevin 24 (4) 641 - 655, december 2017. https://doi.org/10.36045/bbms/1515035013