Answer:
Absolutely. His most famous play "Pygmalion", is a satire (a comedy that makes a political or social comment, usually poking fun or exposing flaws at either/both) that deals with exactly that.
The protagonist Eliza is transformed from a guttersnipe to a "lady" with all the trimmings and fluff associated with being one, especially in her speech; basically Shaw was saying that if it's so easy to transcend through the classes from the lowest to the highest, then being part of the "highest" isn't so special at all.