Genericity of equivariant embeddings

Question

I'd like to ask an equivariant version of this question.

Let $M$ be a closed manifold equipped with the action of a compact Lie group $G$. By the Mostow-Palais embedding theorem, $M$ can be embedded equivariantly into $\mathbb{R}^n$ for some $n$, where $G$ is realized as a subgroup of $GL(n,\mathbb{R})$.

Vague question: Suppose $f:M\to\mathbb{R}^n$ is a continuous $G$-equivariant map. I'd like to know how close $f$ is to an embedding, if we are allowed to make $n$ arbitrarily large.

One attempt at a more precise phrasing:

Question 1: Let $f:M\to\mathbb{R}^n$ be a continuous $G$-equivariant map. Is it true that (assuming $n$ is sufficiently large) for any $\epsilon>0$, there exists a continuous $G$-equivariant embedding $f':M\to\mathbb{R}^n$ such that the image of $f'$ is contained in the $\epsilon$-ball around the image of $f$?

More generally:

Question 2: What if $G$ is a non-compact linear group but the quotient $G\backslash M$ is compact?

Answer 1

Wasserman (Equivariant Differential Topology, Topology, 8(1969), pp. 127-150) proved a generalized Whitney embedding theorem under the assumption that $G$ is compact.

Below you can find the statement of one of his theorems (Corollary 1.10 in the above-mentioned paper).

Def. Let $V$ be a finite dimensional representation of $G$. A $G$-manifold $M$ is subordinate to $V$ if $\forall x \in M \exists U$ invariant neighbourhood of $x$ and equivariant smooth embedding $U\to V^k $ for some $k>0$.

Thm(Wasserman) If $M^n$ is subordinate to $V$, then any equivariant smooth map $f:M\to V^k$ can be approximated $C^r$ and uniformly by an equivariant immersion if $k\geq 2n$ and equivariant 1-1 immersion if $k\geq 2n+1$. If $C$ is a closed subset of $M$, and $f|C$ is an immersion (or embedding)the approximation may be chosen to agree with $f$ on $C$.