Existence of an equivariant Morse function

Question

Let $G$ be a (finite) group and $M$ a $G$ manifold. Now I have a smooth real valued function $f: M\rightarrow R$ with $f(x)=f(g(x)),\, \forall g\in G$. Now in general $f$ will maybe not be a Morse function. However, generally (i.e. ignoring the action of $G$) using Sard's lemma, almost every linear function h will give that f+h is Morse. But now I would like to turn f into an equivariant Morse function. Of course then my choice of linear function is massively constraint and thus this might not be possible. But are there any tricks similar to Sard's lemma in the equivariant case?

Answer 1

This is discussed in

MR0250324 (40 #3563) Wasserman, Arthur G. Equivariant differential topology. Topology 8 1969 127--150.