Let \(M\) be a manifold that is simple homotopy equivalent to a component of the fixed point set of a \(G\)-manifold. The replacement problem for transformation groups asks whether it is possible to realize \(M\) as (a component of) the fixed point set of another \(G\)-manifold that is equivariantly simple homotopy equivalent to the original one. The authors study the replacement problem for locally linear actions of compact Lie groups \(G\) on topological manifolds \(W\). They make the assumption that near a \(1\)-skeleton of the fixed set component \(F\), the action can be identified with a complex \(G\)-bundle whose normal representation is a multiple of \(2\). The main result is that the forgetful map \({S}^G(W)\to { S}(F)\) is split surjective. (Here \({S}(X)\) denotes the structure set of a stratified space \(X\).) In particular, it follows that for such actions, it is always possible to replace the fixed set component by any simple homotopy equivalent (homology) manifold. The authors also correct some computational errors made in some examples, in the addendum to Theorem 0.1 and to Theorem 2.5 in an earlier paper of the first two authors [“Replacement of fixed sets and of their normal representations in transformation groups of manifolds”, Proceedings of a conference in honor of William Browder, Princeton, NJ, USA, March 1994. Princeton, NJ: Princeton University Press. Ann. Math. Stud. 138, 67–109 (1995; Zbl 0930.57031)].