I am wondering if there is a reference for the following:


Let $G$ be a finite group, and suppose that $f\colon M\rightarrow N$ is a continuous and $G$-equivariant map. Here $M$ and $N$ are finite dimensional $G$-manifolds, where $M$ possibly has boundary (I only care when $M$ is compact if that helps). Suppose also that $A\subset M$ is a closed $G$-invariant subset of $M$ on which $f$ is smooth, in that there exists an open neighborhood $U\subset M$ containing $A$ and a smooth equivariant function $h\colon U\rightarrow N$ such that $h\rvert\_A=f\rvert\_A$. Is it true that for any $\epsilon>0$, there exists a smooth equivariant map $f\_\epsilon\colon M\rightarrow N$ such that


1. $\lvert f\_\epsilon(x)-f(x)\rvert<\epsilon$ for all $x\in M$ and
2. $f\_\epsilon\rvert\_A=f\rvert\_A$?



Equivariant Whitney approximation
This is Corollary 1.12 in


*Wasserman, A. G.*, [**Equivariant differential topology**](http://dx.doi.org/10.1016/0040-9383(69)90005-6), Topology 8, 127-150 (1969). [ZBL0215.24702](https://zbmath.org/?q=an:0215.24702).


The proof is essentially the same as the one given by Peter Michor in [his answer](https://mathoverflow.net/a/136765/8103) to [the question linked in the comments by Nick L](https://mathoverflow.net/q/136749/8103).


**Added later:** In fact the above reference does not treat the relative case. A textbook reference for the result in the generality asked for is Theorem 4.2 in Chapter VI of


*Bredon, Glen E.*, Introduction to compact transformation groups, Pure and Applied Mathematics, 46. New York-London: Academic Press. XIII,459 p. $ 21.00 (1972). [ZBL0246.57017](https://zbmath.org/?q=an:0246.57017).


