Equivariant Whitney approximation

Question

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. $f_\epsilon\rvert_A=f\rvert_A$?

Answer 1

This is Corollary 1.12 in

Wasserman, A. G., Equivariant differential topology, Topology 8, 127-150 (1969). ZBL0215.24702.

The proof is essentially the same as the one given by Peter Michor in his answer to the question linked in the comments by Nick L.

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.