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In this paper, we present an experimental study of the metrological stabilization of a solid-state frequency comb for embedded metrology applications. The comb is a passively mode-locked laser diode based on InGaAs/InP Quantum-dash structure emitting optical lines into a 9 nm bandwidth centered at $1.55~\mu \text{m}$ with a repetition rate of 10.09 GHz. The frequency stabilization is achieved by optical injection locking of the comb with an external cavity laser diode referenced onto a metrological frequency standard. One observes the transfer of the spectral purity from the injection laser to the neighbouring modes of the injected one as well as the transfer of stability to the adjacent modes. The measurement of the long term stability highlights a frequency noise with random walk behavior specific of the passive mode locking process. Demonstration of sidebands of the injection laser at the repetition frequency of the comb also makes it possible to propose a transfer mechanism and to consider a complete stabilization of the frequency comb at a metrological stability level. [ABSTRACT FROM AUTHOR]