Kid Gleason.
"Black Sox" was the name given to the 1919 Chicago White Sox whose players (8 of them) were accused of conspiring with gamblers to throw the 1919 World Series. 1919 was Gleason's first as White Sox manager and he led the team through the 1923 season.
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I only found one, Johnny Evers who played and managed the Cubs in 1913 and was the White Sox Manager in 1924.
There is another connection. Don Kessinger played for both teams and was the White Sox manager in 1979.
Robin Ventura, who was a star third baseman for the White Sox from 1989 to 1998, was named manager of the team on October 6, 2011.
Al Lopez.
Lopez managed the White Sox from 1957-1965. He also managed the team for 47 games in 1968 and 17 games in 1969.