Find $f(t)$ such that $\Big(f\circledast f\Big)\times f=g$

Question

If $g$ is a real well-behaved function, find the real function $f$ such that $$\Big(f\circledast f\Big)\times f=g$$ which $\circledast$ is the convolution operator. By taking the Fourier transform of both sides, the problem is equivalent to solving the following functional equation as well. $$f^2\circledast f=g$$