Let $\mathbf{A}$ be a random matrix and $\mathbf{B}$ a matrix composed of constants. Is it possible to take $\mathbf{B}$ out of the expected value: $E\{\mathbf{A}^H\mathbf{BA}\}$.
I tried with simulation and it looks like $E\{\mathbf{A}^H\mathbf{BA}\}\neq \mathbf{B}E\{\mathbf{A}^H\mathbf{A}\}$.
No it is not possible. This is easy to see because if you degenerate your density function for the random matrices so that $A = A_0$ always, then your equation is clearly invalid because matrices do not commute.