  Within the framework of the extended teleparallel gravity, a new class of
boson stars has recently been constructed by introducing the nonminimal
coupling of the scalar field to the torsion scalar. An interesting feature of
these static, spherical, self-gravitating configurations of the massive complex
scalar field is their central region with outwardly increasing energy density,
surrounded by a thick shell within which the joining with the usual
asymptotically Schwarzschild tail takes place. In this work we extend the
original model with the $U(1)$ gauge field and we find that the combined effect
of the charge and coupling of the field to torsion leads to a significant
increase of the maximal mass and the particle number that can be supported
against gravity. We also show that some charged configurations preserve the
property of having the outwardly increasing energy density over the central
region, regardless of the fact that charging the configurations affects the
anisotropy of the pressures in the opposite way relative to that of the
field-to-torsion coupling terms.
