  In the present work, we have computed the shape factors and electron spectra
for the second-forbidden nonunique $\beta^-$-decay transitions of $^{46}$Sc,
and $^{59,60}$Fe in the framework of the nuclear shell model. We have performed
the shell-model calculations of all the involved wave functions in the
$\beta$-decay rate by using the KB3G and GXPF1A interactions in the full $fp$
model space. When compared with the available data, these effective
interactions predict the low-energy spectra and electromagnetic properties of
the involved nuclei quite successfully. This success paves the way for a
reliable computation of the $\beta$-decay properties, and comparison with data.
We use the spectrum-shape method (SSM), including the next-to-leading-order
corrections in the shape factor, in order to compute the electron spectral
shapes as functions of the weak axial coupling constant $g_{\rm A}$. We have
also constrained the value of the relativistic vector matrix element,
$^V\mathcal{M}^{(0)}_{KK-11}$, using the conserved vector-current hypothesis
(CVC) and found that this procedure influences the electron spectral shapes.
Based on the "CVC-inspired" SSM calculations we find that the spectral shapes
of $^{59,60}$Fe depend strongly on the value of $g_{\rm A}$, thus making these
nuclei as perfect test cases of the revised SSM for future $\beta$-decay
experiments that are able to resolve electron spectral shapes.
