I am wondering in which cases the following equality is true: for $X$, $Y$ two random variables and $f$, $g$ two functions,
$$\mathrm{Var}\left[\mathbb{E} ( f(g(X),Y) | X ) \right] = \mathrm{Var}\left[\mathbb{E} ( f(g(X),Y) | g(X) ) \right]. $$
My guess is that the function $g$ should be injective but I did not manage to prove it or to find the proof in literature...
Any help or piece of advice is welcome :)
If $g$ is injective, $E(Z\mid X)=E(Z\mid g(X))$ for every integrable random variable $Z$ hence the result holds.
Otherwise, the result usually fails. Assume for example that $X$ is integrable, $Y=X$, $g:x\mapsto x^2$, $f:(x,y)\mapsto y$, and that the distribution of $X$ is symmetric. Then $f(g(X),Y)=X$ hence $$E(f(g(X),Y)\mid X)=E(X\mid X)=X,\qquad E(f(g(X),Y)\mid g(X))=E(X\mid X^2)=0,$$ which shows that the variances do not coincide in general.