  In this paper we consider the implications that the effect gravitational
memory would have on primordial black holes, within the theoretical context of
$F(R)$ related scalar-tensor theories. As we will demonstrate, under the
assumption that the initial mass of the primordial black hole is such so that
it evaporates today, this can potentially constrain the $F(R)$ related theories
of gravity. We study two scalar-tensor models and discuss the evolution of
primordial black holes created at some initial time $t_f$ in the early
universe. The results between the two models vary significantly which shows us
that, if the effect of gravitational memory is considered valid, some of the
scalar-tensor models and their corresponding $F(R)$ theories must be further
constrained.
