You've already gotten many good answers, but I'll add some thoughts and ideas. In order of importance.
Write Lily as a Person
While diving into the depth of the diagnose and all different ways it manifests, don't forget to write Lily (just like all your other characters) as a person before anything else.
Autism, high-functioning autism, Asperger Syndrome
I'm assuming Lily has "high-functioning autism" or "Asperger syndrome".
However, there are also (I don't know what they want us to call it) low-functioning autism? or classical autism? or just autism? ... needless to say, this is an ongoing debate and hopefully there will be more differing diagnoses in the future, not one umbrella diagnose.
I'm going to talk about Asperger Syndrome.
Autism is a disability
This is the one you need to keep an eye on. If you write Lily as someone with all these symptoms but no real disability, you're not writing a person with Autism, but an eccentric.
In general, health care practitioners won't give a person with no problems a diagnose. Not only because diagnoses might warrant different types of help that cost money, but also because getting a diagnose can be a hard blow and something that can take years to come to terms with.
Lily should suffer in some way from her autism.
Everyone is unique, but there is a diagnose
As some answers mention, every person is unique, but I feel I have to add that there are common traits in Autism... otherwise, there wouldn't be a diagnose... rather these traits may appear in different combinations and strengths in different people, and each person will experience them and deal with them differently.
Autism is usually diagnosed using DSM-5 or ICD-10. And they are a bit different. I've bolded what you might consider adding to Lily's symptoms.
- Problems with social and emotional interactions
- Problems with nonverbal communication
- Problems with developing and maintaining relationships (But I read between the lines you have this one already?)
- Normal early development (language etc. during the first 3 years)
- Problem with social reciprocal interaction
- An unusually intense interest or restrictive, stereotypical patterns of behavior
- No other disability explains the disorder (however, there are so-called comorbidity—see below)
Some personal takes on these criteria:
I think most Aspies will tell you they don't have or don't trust, their gut instincts. We can't make a guestimate about social interactions. Instead, we have to do what I like to call Social Math. Using observation and logical reasoning to figure out what social situations mean. This usually presents as being "slow" to figure these things out, if at all... after all gut reactions are faster than reasoning and logic... unless you're a savant.
It also means a more intellectual approach to emotions, even things like love.
I've seen an example where someone breaks up with an Aspie and can't give a logical reason. This becomes very confusing and upsetting to the aspie. And the ex is just as confused because there's suddenly a request for logic and reason... love isn't logical... right?
Emotions and emotional interaction
Aspies have feelings just like any other person, however, it can sometimes, due to problems reading social situations, be hard for an Aspie to know when to express these feelings or not. It's also harder for an Aspie to read emotions in others, sometimes making them seem emotionless because they can't see what the other person is feeling.
I like to think of both emotions and social interaction as people waving flags, and Aspies as being blind. If people wave the flag really hard, you can understand that some flag waving is going on, but not what color the flag is or what pattern it has.
Unless you do the social math on the situation, what happened before, and what's probably happening now, you're going to have to ask, or be told what's going on, with words.
People like to say 10% of communication is spoken words and 90% is body language. That isn't right. I'd guess about 5% is spoken words, 45% is body language and 50% is what was said before, what everyone knows, what is common and custom in this situation, what's in the Employee Handbook, etc.
Most Aspies will have to fight to not default on what's being said, have to strain to observe body language and do tons of social math to get to the last 50% of what's commonly known as invisible rules.
Unusually intense interest or stereotypical patterns of behavior
An unusually intense interest speaks for itself. I have few, very intense interests. Computers since I was eleven. And writing about since the same age. They haven't changed, and I'm now soon turning 50.
I find myself creating rituals. I wash my hands the same way every time, as well as brushing my teeth and showering. I've had my current oven, a shiny black, flat stovetop that just cries to be kept shiny, for more than ten years and I know with 100% certainty that I have never placed anything on top of it before wiping it off.
I count when pouring water. To "clean old water out of the pipes" I pour to the count of ten with the tap fully opened. When filling my water cooker I count to ten again. To top it off I count to 3 or 5. Filling a pan for cooking, I count to 10 or 15.
When I was younger I counted in more situations, but I guess it's a behavior diminishing with age. And counting is an as good measure as anyone, I guess. However, most of my pans have a measure at the side, and the water cooker has marks inside it. I could just do it on feel... or maybe I can't...
Male vs Female
I'm usually not a fan of male vs female and am of the conviction that these are learned behaviors, but there is a conundrum in autism. Much fewer females than males are diagnosed.
Lily is a female, but I think most of the answers here are about and/or by males (no wonder since there are more males than females with the diagnose). I'm a male.
Look at the link and other research material to find where your female character might differ from a male one. Perhaps the Female Autism Phenotype?
Comorbidity in autism
Your character suffers from autism and kleptomania. This is not at all unusual. Other types of comorbidity (simultaneous diagnoses) could be ADHD, ADD, depression, anxiety and phobias (her problem with crowds?), OCD, etc.
Of course, the number and types of comorbid diagnoses differ from person to person.
Autism usually doesn't appear out of thin air. In many cases diagnosing it in a son will also reveal the father has it. But I think you could see it in siblings and the mother as well... not to mention older generations. My personal experience; it abounds, in history and present time.
However, other diagnoses don’t have to be autism. It could be any combination of diagnoses not limited to the list under comorbidity above.
This will give a character with autism the possibility of emotional wounds... having parents with diagnoses... perhaps dealing with them through substance abuse or failing at parenting.
That Lily was abandoned is extreme but most possible. It happened to one of my grandparents...
Here are other things that are common symptoms in Aspies.
People with autism have a hard time doing what is commonly called mentalization. It's seldom natural for an Aspie to think about what other people are thinking, and I've found that even after you know this, you have to keep reminding yourself to try. Many misunderstandings come from not realizing other people don't know or share your thoughts or interests.
As an author, I spend lots of time thinking about what my characters think and feel, and what other people think and feel, but when I am in a situation that requires mentalization, I still have a hard time getting it right. It's easy when you can spend hours and days doing it, even months, but when you need to do it in the moment... it sucks!
Uneven talent profile
Aspies usually have an uneven talent profile. This means that while they can be geniuses in some respects (e.g. logic, math, pattern recognition, endurance, and meticulousness) they can be outright "idiots" in other areas (e.g. social interactions, house cleaning, romantic relations.)
This surprises many people without the diagnose. The common assumption is that knowing a person's competence in one area will give a good estimate of said person's competence in another.
The fact that Lily makes a living by stealing reveals that she is probably pretty good at doing the social math required to navigate all the straits such a life contains. This, in turn, makes her a pretty intellectually capable person. And still, she's decided to use all that mental capacity to steal. Likely she's a (bit of a/total) dimwit when it comes to long term planning, or she'd used that mental capacity to land a proper job.
But I've seen similar behavior in Aspies as well. Smart people with super educations, that just aren't able or willing to do what they need to land a job.