I'm looking for a reference for the following:


Suppose that $G$ is a finite group, that $M$ is a smooth $G$-manifold, and that $A\subseteq M$ is a closed $G$-invariant subspace of $M$ such that the action on $M\setminus A$ is free. Suppose also that $G$ acts on $\mathbb{R}^n$. If $f\colon M\rightarrow \mathbb{R}^n$ is a smooth $G$-equivariant map which is transverse to zero on $A$, then for any $\epsilon>0$ there exists a smooth $G$-equivariant map $f'\colon M\rightarrow \mathbb{R}^n$ such that


(1) $|f'(x)-f(x)|<\epsilon$ for all $x\in M$,


(2) $f'$ is transverse to zero, and


(3) $f'|\_A=f|\_A$.



Relative equivariant Thom transversality
See Prop 2.2 of ON THE GROUPS JO(G), Chung-Nim Lee and Arthur Wasserman, Memoirs of the American Mathematical Society Number 159


