Set them free, set them anywhere and they will feel at home in a few minutes. Extroversion or extraversion is an essential personality trait. People who are extrovert are stars of parties and gatherings, they are the glue that makes a party solid.
Extrovert kids freely move from one set of arms to another and are playful even with strangers. Their curiosity needs to be addressed almost the minute it pops up. This is because their next minute is for the next curiosity. A very high degree of engagement is demanded by extrovert kids at home and at familiar places, however in unfamiliar surrounding they are the center of attention.
As parents, we need to make sure we are never emotionally disconnected, restrictive or authoritarian with them. This is where it hurts them most. Leadership and initiative come naturally to an extrovert child. As parents, we need to give them everyday opportunities to hone their skills and show them off. Showing them off gives them gratification and a sense of achievement.
Be by their side, despite being extrovert kids, they need parenting.
PS: Extraversion and Extroversion are two different spellings, both acceptable but extrovert/extroversion is now almost default set of spellings.