  The scheme and gauge dependence of the factorization property of the RG
$\beta$-function in the $SU(N_c)$ QCD generalized Crewther relation (GCR),
which connects the non-singlet contributions to the Adler and Bjorken polarized
sum rule functions, is investigated at the $\mathcal{O}(a^4_s)$ level. In the
gauge-invariant $\rm{\overline{MS}}$-scheme this property holds at least at
this order. To study whether this property is true in all gauge-invariant
schemes, we consider the $\rm{MS}$-like schemes in QCD and the QED-limit of the
GCR in the $\rm{\overline{MS}}$-scheme and in the $\rm{MOM}$ and the $\rm{OS}$
schemes. In these schemes we confirm the existence of the $\beta$-function
factorization in the QCD and QED variants of the GCR. The problem of the
possible $\beta$-factorization in the gauge-dependent renormalization schemes
in QCD is studied. We consider the gauge non-invariant $\rm{mMOM}$ and
$\rm{MOMgggg}$-schemes and demonstrate that in the $\rm{mMOM}$ scheme at the
$\mathcal{O}(a^3_s)$ level the $\beta$-factorization is valid for three values
of the gauge parameter $\xi$ only, namely for $\xi=-3, -1$ and $\xi=0$. In the
$\mathcal{O}(a^4_s)$ order of PT it remains valid only for case of the Landau
gauge $\xi=0$.
  The consideration of these two schemes for the QCD GCR allows us to conclude
that the factorization of RG $\beta$-function will always be implemented in any
$\rm{MOM}$-like schemes with linear covariant gauge at $\xi=0$ and $\xi=-3$ at
the $\mathcal{O}(a^3_s)$ level. It is demonstrated that if factorization
property for the $\rm{MS}$-like schemes is true in all orders of PT, as
theoretically indicated, then the factorization will also occur in the
arbitrary $\rm{MOM}$-like scheme in the Landau gauge in all orders of PT as
well.
