Don't feel it has to be a "standard" social activity like footy or scouts or ballet lessons etc. Perhaps something smaller scale like dog walking with another Mum at the local shelter would be a better start, aiming to build up to the standard stuff over time?
My son's 1/2 bro has the competive thing going on and adored karate! My own child who isn't bothered what others do as he's too wrapped up in his own world but is OTT on the impulsive front has taken to akido like a duck to water. Best still the Akido is REALLY helping his difficulties with positional intructions and seqencing language difficulties (I don't understand how but in the abscence of any meaningful salt input you give thanks for what works ya know?)
Selecting a good instructor is crucial though - my lad 's instructor is huge on consideration for your sparring partner (no injuries that way!) and self-discipline. It's good for asd kids to have to consider another person's feelings etc in any context. Akido makes it necessary as you can't improve without co-operation. However it's only focusing on one other individual - your partner and the instructor at intervals, not a whole football team or whatever iykwim. He's also learnt breathing techniques to help him calm himself down at other times.
I'd really rate martial arts but the specific choice from karate through to tai chi or boxing should be selected based on your child's own personaility as they all have their on "flavour" and some are more or much less aggressive than others. The instructor is critical too. DS's are superb and have a lot of experience with adhd, deaf and asd kids, others have no experience of sen and are a bit gung ho and macho for my tastes. Watch a few sessions and then let your child watch a session before enrolling them.
Ds is also following in his Dad's footsteps with Chess - lots of areas have kids chess groups and DS will be enrolling in September. That's a non- physical "social" activity. Dungeons and Dragons or World or War craft groups are popular with teens as are LARP groups for other ideas.
Riding for the disabled suits some kids to a tee, or joining a city farm group to help clean out the animals. Volunteers for city farms tend to be mixed ages but a few older role models can sometimes be better for our kids than a hoard of same age kids. Sunday schools are good, bad and ugly depending on the group. Or if all else fails a friendly dog can break the ice on walks with Mum in the park.