  The semi-analytical expression for the forth coefficient of 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 three-loop perturbative approximation for the QCD static potential,
evaluated in the $\rm{\overline{MS}}$-scheme. The importance of getting more
detailed expressions for the $n_f$-independent three-loop contribution to the
static potential,obtained at present by two groups, is emphasised. The
comparison of the numerical structure of the four-loop approximations for the
RG $\beta$- function of QCD in the gauge-independent ${\rm{V}}$- and
$\rm{\overline{MS}}$-schemes and in the minimal MOM scheme in the Landau gauge
are presented. Considering the limit of QED with $N$-types of leptons we
discover that the $\beta^{\rm{V}}$-function is starting to differ from the
Gell-Mann--Low function $\Psi(\alpha_{\rm{MOM}})$ at the level of the
forth-order perturbative corrections, receiving the proportional to $N^2$
additional term. Taking this feature into account, we propose to consider the
$\beta^{\rm{V}}$-function as the most theoretically substantiated analog of the
Gell-Man--Low function in QCD.
