The equivariant Atiyah-Jänich theorem is an isomorphism $$ [X,F]_G \cong K_G^0(X), $$ where $G$ is a compact Lie group, $X$ is a compact $G$-manifold, $F$ is the space of Fredholm operators on a certain $G$-Hilbert space, $[-,-]_G$ denotes $G$-homotopy classes of $G$-maps, and $K_G^0(X)$ is the $G$-equivariant K-theory of $X$, defined as the Grothendieck group of $G$-equivariant vector bundles over $X$.

The standard reference for the equivariant Atiyah-Jänich theorem (see, e.g. this question) seems to be Matumoto, T., Equivariant K-theory and Fredholm operators. J. Fac. Sci. Univ. Tokyo Sect. I A Math. 18 1971 109–125.

The proof given there proceeds like Atiyah's proof of the original theorem, it constructs an index map $[X,F]_G \to K_G^0(X)$, shows surjectivity, identifies the kernel with a space $[X,GL]_G$ of $G$-homotopy classes, and then proves the $G$-equivariant contractibility of $GL$, so that the kernel vanishes.

Now, while the $G$-equivariant contractibility of $GL$ is treated carefully in all details, the construction of the index map doesn't seem to be treated at all (,,...will be expected to give an element...``). In particular, none of the difficulties of Atiyah's proof is mentioned, e.g. finding a closed subspace complementary to the kernels, etc.

Also, a proof of surjectivity of the index map is only scetched in one sentence on page 3.

So, my question is: is there a reference for the $G$-equivariant index map?

I have looked into Segal's paper on equivariant K-theory and have also checked the literature given in the answers to this question.