It is normal to extend fixed-point amplitude amplification to an oblivious version, i.e., $1 - (1-e^{i \beta})|t\rangle \langle t | \rightarrow 1 - (1-e^{i \beta}) 1 \otimes |0\rangle \langle 0|$, and one uses an additional ancilla to apply a phase to the target state $|t\rangle$.
However typically the other reflection, $1 - (1-e^{-i\alpha}) |0\rangle \langle 0|$ is about the all-zero initial state. But suppose you are given an unknown initial state, $|\psi_0 \rangle$. You know you can construct a unitary $U$ that does $$ U |\psi_0\rangle |0\rangle = \sqrt{a}|A_0\rangle |0\rangle + \sqrt{1-a}|A_1\rangle |1\rangle, $$ and so could amplify the probability of $|A_{0}\rangle |0\rangle$ to 1, but only if you could implement $1 - (1-e^{-i\alpha})|\psi_0\rangle\langle \psi_0 | \otimes |0\rangle \langle 0|$, and you don't know how $|\psi_{0}\rangle$ came about. Is it possible instead to construct a unitary $U'$ that will give you $$ U' |\psi_0\rangle |0\rangle = \sqrt{a}|A_0\rangle |0\rangle + \sqrt{1-a}|A_1\rangle |1\rangle, $$ while simultaneously only using $1 - (1-e^{-i\alpha})1\otimes |0\rangle \langle 0|$ for the initial state reflection during amplitude amplification? It is not clear to me how one would construct a $U'$ that moves between the two bases $|A_0\rangle |0\rangle$, $|A_1\rangle |1\rangle$, and $|\psi_0\rangle |0\rangle$, $|\psi_{1}\rangle |1\rangle$, for some $|\psi_1 \rangle |1\rangle$.
