The spacetime in the interior of a black hole can be described by an homogeneous line element, for which the Einstein–Hilbert action reduces to a one-dimensional mechanical model. We have shown in Geiller et al (2021 SciPost Phys. 10 022) that this model exhibits a symmetry under the (2 + 1)-dimensional Poincaré group. Here we extend the Poincaré transformations to the infinite-dimensional BMS$_{3}$ group. Although the black hole model is not invariant under those extended transformations, we can write it as a geometric action for BMS$_{3}$, where the configuration space variables are elements of the algebra and the equations of motion transform as coadjoint vectors. The BMS$_{3}$ symmetry breaks down to its Poincaré subgroup, which arises as the stabilizer of the vacuum orbit. This symmetry breaking is analogous to what happens with the Schwarzian action in AdS$_{2}$ JT gravity, although in the present case there is no direct interpretation in terms of boundary symmetries. This observation, together with the fact that other lower-dimensional gravitational models (such as the BTZ black hole) possess the same broken BMS$_{3}$ symmetries, provides yet another illustration of the ubiquitous role played by this group.