  Primordial black holes (PBHs) forming out of the collapse of enhanced
cosmological perturbations provide access to the early Universe through their
associated observational signatures. In particular, enhanced cosmological
perturbations collapsing to form PBHs are responsible for the generation of a
stochastic gravitational-wave background (SGWB) induced by second-order
gravitational interactions, usually called scalar induced gravitational waves
(SIGWs). This SGWB is sensitive to the underlying gravitational theory; hence
it can be used as a novel tool to test the standard paradigm of gravity and
constrain possible deviations from general relativity. In this work, we study
the aforementioned GW signal within modified teleparallel gravity theories,
developing a formalism for the derivation of the GW spectral abundance within
any form of gravitational action. At the end, working within viable $f(T,\phi)$
models without matter-gravity couplings, and accounting for the effect of
mono-parametric $f(T)$ gravity at the level of the source and the propagation
of the tensor perturbations, we show that the respective GW signal is
indistinguishable from that within GR. Interestingly, we find that in order to
break the degeneracy between different $f(T)$ theories through the portal of
SIGWs one should necessarily consider non-minimal matter-gravity couplings at
the level of the gravitational action.
