$f\circ(g\circ g)=(g\circ g)\circ f$; $f\circ g\ne g\circ f$

Question

Let $f\circ(g\circ g)=(g\circ g)\circ f$ but not equal for $f\circ g\ne g\circ f$. What are possible $f$ and $g$?

Answer 1

$f(x)=x^2$ and $g(x)=-x$ seem to work.

Answer 2

Choose: $g(x) =1/x$; or any $g$ with $g(g(x))= x = id(x).$

Now find $f$ that does not commute with $g$.