The Fourier transform $\mathcal F$ is defined by $$Ff(\xi)=\int_{-\infty}^\infty f(t)e^{-i\xi t}dt$$
Generally, the domain of $\mathcal F$ is $L^1$ since otherwise the integral isn't defined, and the codomain of $\mathcal F$ is $C_0$, which is possible by Riemann-Lebesgue Lemma.
On the other hand, Fourier inverse transform $\mathcal F^{-1}$ is defined by $$\mathcal F^{-1}f(t)=\dfrac{1}{2\pi}\int_{-\infty}^\infty f(\xi)e^{i\xi t}d\xi$$ (Note that $\mathcal F^{-1}$ doesn't mean the inverse mapping of $\mathcal F$.)
I'm wondering what the domain and codomain of Fourier inverse transform is. I searched but I do not find.
Maybe, similarily to Fourier transform, the domain of $\mathcal F^{-1}$ is $L^1$ since otherwise the integral isn't defined. (right?)
And, generally, what is the codomain of Fourier inverse transform ?
The question isn't quite well-formed because there are many valid choices of domain for $\mathcal{F}$ and $\mathcal{F}^{-1}$. It depends on what specifically you're trying to do.
As you have observed, the integral formulas for $\mathcal{F}$ and $\mathcal{F}^{-1}$ are well-defined for $L^1$ functions, but not necessarily so for other functions. With domain $L^1$, both $\mathcal{F}$ and $\mathcal{F}^{-1}$ have codomains $C_0$, the space of continuous functions vanishing at infinity. Of course, this means that these are not true inverses of each other in this context.
But it is also possible to define $\mathcal{F}$ and $\mathcal{F}^{-1}$ on other domains. For example, $\mathcal{F}$ can be extended to $L^2$ via continuous approximation from the dense subspace $L^1\cap L^2$. In this case $\mathcal{F}$ is no longer defined solely through the integral formula; instead it is defined as a limit. Under this definition, $\mathcal{F}$ can be defined as a map $L^2\to L^2$, and one can also show that an analogous definition of $\mathcal{F}^{-1}:L^2\to L^2$ serves as its inverse.
One can also work on a smaller space, such as the space of Schwartz functions $\mathcal{S}$. In that case it is easy to show that the integral definition of $\mathcal{F}$ is an automorphism of Schwartz space and the integral definition of $\mathcal{F}^{-1}$ is its inverse. One can then generalize this to extend $\mathcal{F}$ to to the space $\mathcal{S}'$ of tempered distributions, which is a much larger space than $L^1$. This turns out to be an isomorphism as well, with inverse $\mathcal{F}^{-1}$ (also suitably extended to a map $\mathcal{S}'\to\mathcal{S}'$.