  Context. The CoRoT and Kepler missions have paved the way for synergies
between exoplanetology and asteroseismology. The use of seismic data helps
providing stringent constraints on the stellar properties which directly impact
the results of planetary studies. Amongst the most interesting planetary
systems discovered by Kepler, Kepler-444 is unique by the quality of its
seismic and classical stellar constraints. Its magnitude, age and the presence
of 5 small-sized planets orbiting this target makes it an exceptional testbed
for exoplanetology. Aims. We aim at providing a detailed characterization of
Kepler-444, focusing on the dependency of the results on variations of key
ingredients of the theoretical stellar models. This thorough study will serve
as a basis for future investigations of the planetary evolution of the system
orbiting Kepler-444. Methods. We use local and global minimization techniques
to study the internal structure of the exoplanet-host star Kepler-444. We
combine seismic observations from the Kepler mission, Gaia DR2 data and revised
spectroscopic parameters to precisely constrain its internal structure and
evolution. Results. We provide updated robust and precise determinations of the
fundamental parameters of Kepler-444 and demonstrate that this low-mass star
bore a convective core during a significant portion of its life on the
main-sequence. Using seismic data, we are able to estimate the lifetime of the
convective core to approximately 8 Gyr out of the 11 Gyr of the evolution of
Kepler-444. The revised stellar parameters found by our thorough study are M =
0.754 +- 0.03M_Sun , R = 0.753 +- 0.01R_Sun , Age = 11 +- 1 Gy.
