Should mean wages increase with inflation? Should mean wages increase with inflation? 
If not, why? Has this profound consequences in the redistribution of wealth?
(not an expert in economics...I apologise is the question is too naif)
Should mean wages increase with inflation? If not, why?


No. Inflation is not the only factor that determines wages. Inflation has to do with the nominal value of wages, but these also entail components pertaining to the real value of work.

Here are three examples that tend to push mean wages down,

(1) technology, to the extent that it replaces human labor;

(2) foreign markets may lure high skilled workers (or, for what matters, workers with the highest salaries) to emigrate, causing -caeteris paribus- mean wages to decrease;

(3) public policy (whether it stems from unions or from policies aimed at reducing the number of unemployed) might prompt -otherwise unneeded- entry of more people in the labor market.

Other factors would have the effect of pushing mean wages up. For instance, the converse of (2) (that is, emigration only of workers earning the lowest salaries), unavailability of workers with high productivity, and so forth.


  Has this profound consequences in the redistribution of wealth?


That depends on the particularities of the economy being analyzed: what fiscal policy is implemented; whether households own significant portions of capital; the sensitivity and preponderance of different sectors in the economy; etc.