  We consider a semi-infinite one-dimensional phase-change material with two
unknown constant thermal coefficients among the latent heat per unit mass, the
specific heat, the mass density and the thermal conductivity. Aiming at the
determination of them, we consider an inverse one-phase Stefan problem with an
over-specified condition at the fixed boundary and a known evolution for the
moving boundary. We assume that the phase-change process presents latent-heat
memory effects by considering a fractional time derivative of order $\alpha$
($0<\alpha<1$) in the Caputo sense and a sharp front model for the interface.
According to the choice of the unknown thermal coefficients, six inverse
fractional Stefan problems arise. For each of them, we determine necessary and
sufficient conditions on data to obtain the existence and uniqueness of a
solution of similarity type. Moreover, we present explicit expressions for the
temperature and the unknown thermal coefficients. Finally, we show that the
results for the classical statement of this problem, associated with
$\alpha=1$, are obtained through the fractional model when $\alpha\to 1^-$.
