Indeed when we compare the inequality present in different industrialized nations with varying degrees of unionization, nations with stronger unions seem to have lower inequality despite experiencing an increase in automation. As stated in an IMF report [1]:
Traditional explanations for the rise of inequality in advanced economies are skill-biased technological change and globalization, which have increased the relative demand for skilled workers, benefiting top earners relative to average earners. But technology and globalization foster economic growth, and there is little policymakers can or are willing to do to reverse these trends. Moreover, while high-income countries have been similarly affected by technological change and globalization, inequality in these economies has risen at different speeds and magnitudes.
So perhaps the combination of automation together with weakened unions is responsible for inequality in industrialized nations?
[1] https://www.imf.org/external/pubs/ft/fandd/2015/03/pdf/jaumo...