Electromagnetically induced transparency allows for the controllable change of absorption properties, which can be exploited in a number of applications including optical quantum memory. In this paper, we present a study of the electromagnetically induced transparency in a 167Er:7LiYF4 crystal at low magnetic fields and ultra-low temperatures. The experimental measurement scheme employs an optical vector network analysis that provides high precision measurement of amplitude, phase and group delay and paves the way towards full on-chip integration of optical quantum memory setups. We found that sub-Kelvin temperatures are the necessary requirement for observing electromagnetically induced transparency in this crystal at low fields. A good agreement between theory and experiment is achieved by taking into account the phonon bottleneck effect.