I use the definition of Steenrod's equivariant cohomology given in chapter V of the book
N. E. Steenrod. Cohomology Operations. No. 50 in Annals of Mathematics Studies. Princeton University Press, 1969. Link to PDF
Steenrod uses $G$-complexes with $G$ in most cases a finite group. I try to show that there is an isomorphism between $H_G^n(C_\ast^{sing}(X);A)$ and $H_G^n(C_\ast^{CW}(X);A)$ with $A$ a $G$-module and $X$ a $G$-complex.
Is there any source which already states a claim like this or which gives a hint?