  Pulsating low-mass white dwarf stars are white dwarfs with stellar masses
between 0.30~M$_{\odot}$ and 0.45~M$_{\odot}$ that show photometric variability
due to gravity-mode pulsations. Within this mass range, they can harbour both a
helium- and hybrid-core, depending if the progenitor experienced helium-core
burning during the pre-white dwarf evolution. SDSS J115219.99$+$024814.4 is an
eclipsing binary system where both components are low-mass white dwarfs, with
stellar masses of 0.362$\pm$0.014~M$_{\odot}$ and 0.325$\pm$0.013~M$_{\odot}$.
In particular, the less massive component is a pulsating star, showing at least
three pulsation periods of $\sim$1314 s, $\sim$1069 s and $\sim$582.9 s. This
opens the way to use asteroseismology as a tool to uncover its inner chemical
structure, in combination with the information obtained using the light-curve
modelling of the eclipses. To this end, using binary evolutionary models
leading to helium- and hybrid-core white dwarfs, we compute adiabatic
pulsations for $\ell=1$ and $\ell=2$ gravity modes with \texttt{Gyre}. We found
that the pulsating component of the SDSS J115219.99$+$024814.4 system must have
a hydrogen envelope thinner that the value obtained from binary evolution
computations, independently of the inner composition. Finally, from our
asteroseismological study, we find a best fit model characterised by T$_{\rm e
ff}=10\, 917$ K, M=0.338~M$_{\odot}$, M$_{\rm H}=10^{-6}$~M$_{\odot}$ with the
inner composition of a hybrid WD.
