  We study the evolution of cosmological perturbations around a homogeneous and
isotropic background in the framework of the non-minimal torsion-matter
coupling extension of $f(T)$ gravity. We are concerned with the effects of the
non-minimal coupling term on the growth of matter overdensities. Under the
quasi-static approximation, we derive the equation which governs the evolution
of matter density perturbations, and it is shown that the effective
gravitational coupling 'constant' acquires an additional contribution due to
the non-minimal matter-torsion coupling term. In this way, this result
generalizes those previously obtained for the growth of matter overdensities in
the case of minimal $f(T)$ gravity. In order to get a feeling of our results we
apply them to the important case of a power-law coupling function, which we
assume to be the responsible for the late-time accelerated expansion in the
dark energy regime. Thereby, analytic solutions for the matter density
perturbation equation in the regime of dark matter dominance and the dark
energy epoch are obtained, along with a complete numerical integration of this
equation. In particular, we show that this model predicts a growth index larger
than those obtained for $\Lambda$CDM model, indicating therefore a smaller
growth rate. Concomitantly, we show that the model at hand is potentially
capable in alleviating the existing $\sigma_{8}$-tension, being that it can
provide us a $f\sigma_{8}$ prediction which is $\sim 4-5$ per cent below the
respective prediction of concordance model.
