Matrix Equation $A^* B A = C $ solved for $A$

Question

Is there a standardized way to solve

$A^* B A = C $

for $A$ if $A$ is a complex and square matrice, and $B$ and $C$ are real-valued and square matrices. $A^*$ is the conjugate transpose of $A$.

Is there a special name for such kinds of equations?

Edit:

Information that might help: The Matrix B and C are conjugate transpose auto-correlation matrices

$B^* = b \cdot b^* \\ C^* = c \cdot c^* $

Answer 1

If $B,C$ are positive definite self-adjoint matrices, then they have unique Cholesky decomposition $B=L_BL^*_B$, $C=L_CL_C^*$. Substituting into the equation $$A^*L_BL_B^*A=L_CL_C^*$$ $$\therefore L_B^*A=UL_C^*$$ $$A=L_B^{-*}UL_C^*$$ where $U$ is any unitary matrix.