Let $G$ be a topological group and $X$, $Y$ be $G$-spaces. Suppose there is a continuous map $F:X\to Y$ which is $G$-equivariant up to homotopy, namely the two maps $G\times X\to Y$, $(g,x)\mapsto F(gx)$ and $(g,x)\mapsto gF(x)$ are homotopic.
Then does $F$ induce a map in $G$-equivariant homology (in the sense of Borel construction), or maybe something well-defined up to homotopy?
If not, are there any conditions that make the statement true, or are there similar results?