S.O.
You've gotten a lot of good advice. I agree on not pushing the play dates often. A few every once in a while is good for an introvert. An introvert is not necessarily a shy, quiet person, but when give some free time introverts recharge in solo activities. Extroverts recharge when they are around people.
2 of my 3 kids are introverts. They still like doing things with their friends and going places, but they took a little longer to develop socially. When they were preschoolers, they'd play NEXT to each other in the same room. Sometimes with each other or they'd do something together...but many times one building legos right next to the one playing with trains.
Their social strengths weren't visible until about age 11 or 12. That's the schedule some people are on.