Relating two similar matrices

Two $n \times n$ matrices $Q_1$ and $Q_2$ are diagonalizable and have the same eigenvalues, thus they are said to be similar , and there exists a matrix $A$ such that

$Q_2 = A^{-1} Q_1 A$

Suppose $Q_2 = {A_1}^{-1} Q_1 A_1 = {A_2}^{-1} Q_1 A_2$ , is it true that $A_2$ must be a (non-zero) multiple of $A_1$ ?