  The neutrino minimal standard model ($\nu$MSM) has been tightly constrained
in the recent years, either from dark matter (DM) production or from X-ray and
small-scale observations. However, current bounds on sterile neutrino DM can be
significantly modified when considering a $\nu$MSM extension, in which the DM
candidates interact via a massive (axial) vector field. In particular, standard
production mechanisms in the early Universe can be affected through the decay
of such a massive mediator. We perform an indirect detection analysis to study
how the $\nu$MSM parameter-space constraints are affected by said interactions.
We compute the X-ray fluxes considering a DM profile that self-consistently
accounts for the particle physics model by using an updated version of the
Ruffini-Arg\"uelles-Rueda (RAR) fermionic ("ino") model, instead of
phenomenological profiles such as the Navarro-Frenk-White (NFW) distribution.
We show that the RAR profile accounting for interacting DM, is compatible with
measurements of the Galaxy rotation curve and constraints on the DM
self-interacting cross section from the Bullet cluster. A new analysis of the
X-ray NuSTAR data in the central parsec of the Milky Way, is here performed to
derive constraints on the self-interacting sterile neutrino parameter-space.
Such constraints are stronger than those obtained with commonly used DM
profiles, due to the dense DM core characteristic of the RAR profiles.
