A Sjolin–Soria–Antonov type extrapolation theorem for locally compact $$\sigma$$ -compact non-discrete Hausdorff groups is proved. Applying this result it is shown that the Fourier series with respect to the Vilenkin orthonormal systems on the Vilenkin groups of bounded type converge almost everywhere for functions from the class $$L {\rm log}^{+} L {\rm log}^{+} {\rm log}^{+} {\rm log}^{+} L$$ .