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Following the methods of G. Gát, in this work we prove the a.e convergence of the subsequence(   σm   n    2    M   nf   )   n$\begin{array}{} (\sigma_{\frac{m_{n}}{2}M_{n}}f)_{n} \end{array}$, for every integrable function  f  on unbounded Vilenkin groups, such that the sequence ( m n  )  n   contains infinitely many even terms satisfying the estimateln  ⁡   m   k  −  1    ln  ⁡   m   km   k=  O  (  1  )  .$\begin{array}{} \frac{\ln m_{k-1}\ln m_{k}}{m_{k}}=O(1). \end{array} $

Almost everywhere convergence of some subsequences of Fejér means for integrable functions on some unbounded Vilenkin groups