Suppose for any real number $a$, we have a function $f_a:\mathbb R \to \mathbb R$ or  such that $f_a(x)$ is monotonically strictly increasing in $x$ and hence invertible on its image. We also assume continuity in $a$.

We want to find conditions on $f$ such that there exist $a_1,a_2:\mathbb R \to \mathbb R$ such that:

$$\exists F:\mathbb R^2 \to \mathbb R, \quad f_a(F(x,y))=f_{a_1(a)}(x)\cdot f_{a_2(a)}(y)$$
Or in other words, $f^{-1}_a(f_{a_1(a)}(x)\cdot f_{a_2(a)}(y))$ does not depend on $a$.

Is this a known problem? There are a number of $f$ that satisfy this property, such as $f_a(x)=a\cdot x$ and $f_a(x)=e^{a\cdot x}$, but is there a general characterization, or at least a relatively tight set of necessary conditions on $f$?