  Primordial black hole (PBH) fluctuations can induce a stochastic
gravitational wave background at second order, and since this procedure is
sensitive to the underlying gravitational theory it can be used as a novel tool
to test general relativity and extract constraints on possible modified gravity
deviations. We apply this formalism in the framework of $f(T)$ gravity,
considering three viable mono-parametric models. In particular, we investigate
the induced modifications at the level of the gravitational-wave source, which
is encoded in terms of the power spectrum of the PBH gravitational potential,
as well as at the level of their propagation, described in terms of the Green
function which quantifies the propagator of the tensor perturbations. We find
that, within the observationally allowed range of the $f(T)$ model-parameters,
the obtained deviations from general relativity, both at the levels of source
and propagation, are practically negligible. Hence, we conclude that realistic
and viable $f(T)$ theories can safely pass the primordial black hole
constraints, which may offer an additional argument in their favor.
