  The semi-analytical $O(\alpha_s^4)$ expression for the renormalization group
$\beta$-function in the ${\rm{V}}$-scheme is obtained in the case of the
$SU(N_c)$ gauge group. In the process of calculations we use the existing
information about the three-loop perturbative approximation for the QCD static
potential, evaluated in the $\rm{\overline{MS}}$-scheme. The comparison of the
numerical values of the third and fourth coefficients for the QCD RG $\beta$-
functions in the gauge-independent ${\rm{V}}$- and $\rm{\overline{MS}}$-schemes
and in minimal MOM scheme in the the Landau gauge is presented. The
phenomenologically-oriented comparisons for the coefficients of $O(\alpha_s^4)$
expression for the $e^+e^-$-annihilation R-ratio in these schemes are
presented. It is shown, that taking into account of these QCD contributions are
of vital importance and lead to the drastic decrease of the scheme-dependence
ambiguities of the fourth-order perturbative QCD approximations for the
$e^+e^-$ annihilation R-ratio for the number of active flavours,$n_f=5$ in
particular. We demonstrate that in the case of QED with $N$-types of leptons
the coefficients of the $\beta^{\rm{V}}$-function are closely related to the
ones of the Gell-Mann--Low $\Psi$-function and emphasise that they start to
differ from each other at the fourth order due to the appearance of the extra
$N^2$-contribution in the V-scheme. The source of this extra correction is
clarified. The general all-order QED relations between the coefficients of the
$\beta^{\rm{V}}$- and $\Psi$-functions are discussed.
