  The famous Carleson-Hunt theorem has been in focus of interest for a long
time. This theorem concerns convergence almost everywhere of Fourier series of
$f\in L_p$ functions for $1<p\leq \infty.$ Kolmogorov constructed a function
$f\in L_1$ such that the partial sums of Fourier series diverge everywhere. On
the other hand, we have boundedness result for F\'ejer means for all $1\leq
p\leq \infty$. Similar results are proved for the partial sums and F\'ejer
means of Vilenkin-Fourier series. But also here it appears the questions what
happens on any subset $E$ of measure zero, can we even have a function which
diverge there? We contribute with a new result concerning this question and
prove by the concrete construction that for any set $E$ of measure zero there
exists a function $f\in L_p(G_m) (1\leq p<\infty)$ such that the F\'ejer means
with respect to Vilenkin systems diverge on this set, which follows similar
result for the partial sums. The key is to use new constructions of Vilenkin
polynomials, which was introduced in \cite{PTW2}. In fact, the theorem we prove
follows from the general result of \cite{Kar}, but we provide an alternative
approach and the constructed function in our proof has a simple explicit
representation.
