Sufficient conditions are given in order that a sequence of linear operators $L_n(\Lambda,\cdot)$ defined by $$ L_n(\Lambda,f):=\sum_{k=0}^n\lambda_{nk}\hat{f}(k)\chi_k\quad (n\in {N}_0),\quad \hat{f}(k):= \int_G\limits f\overline{\chi}_k\ (k\in N_0), $$ converges in $L^q$- norm to identity, where $f\in L^q(G)$, $q\in [1,\infty]$, $\lambda_{n0}=1\;(\forall n\in {N}_0)$, $\lambda_{nk}=0\;(\forall k\gt n,\forall n\in {N}_0)$ and $G$ is a general Vilenkin group. In case of bounded Vilenkin groups, our result coincides with an earlier result of Blyumin.