I need to compute the derivative of
$$ (m^T \otimes a^H) [(m^* \otimes I)(m^T \otimes I)]^{-1} (m^* \otimes a) $$
with respect to $ m $, which is a complex-valued vector. Moreover, $ a $ is also a complex vector and $ I $ is the identity matrix. Finally, $ (\cdot)^T, (\cdot)^H, (\cdot)^* $ denote the transpose, Hermitian transpose, and conjugate, respectively.
There exists a similar question in Derivative of Nested Matrix Quadratic Form that uses differentials. However, the Kronecker product complicates my derivation.