I don't have the same situation as yours but your sons behaviour depends on his training. They need to have a routine every day, from the time they wake up to whatever they do during the day and up to the time they go to bed.
ASD kids love routines. Even the mild cases, they love it.
And please remember, there is NO CURE for ASD. They get better if you train them properly or according to what you think is right for them. You have to train them what is right and wrong in every thing that they do and in every situation too. They can only see black and white, not in between. It is either a "Yes" or a "No" to them.
If you train them properly, they will improve as time goes by. They will improve in the way they behave towards people around them like the family members and also towards strangers. That also includes the way they behave in school.
In your step son's case, you must be doing the right thing. But when he goes back to his mother, maybe she is not doing it right!
Think of what you do to him when he is in your place from the time he wakes up to the time he goes to bed. Then ask his mom what does she do to him from the time he wakes up to the time he goes to bed. Then compare the two and see what's the difference.
There has to be a difference between how you deal with him in your place and how his mom deals with him in her place.
Then, you can try and tell her what to do as an experiment and see what happens. She has to try whatever you are doing right for her own sake.
How old is your step son,BTW? And what is his main problem in his disability?
My son's problem is mainly academic and social behaviour. He is much better at the social behaviour now but we still have to teach him things every now and then. Academically, he cannot do things on his own most of the time. He needs help a lot with his school work and the school provides him teacher- aide in the subjects that is very difficult for him to understand. I had to ask the school to do this. And I have to keep reminding them every now and then. My son is now 15 yrs old and he is going in a mainstream high school.
Just remember that ASD kids love routine. If you follow their routine everyday, they are alright. If they are well trained enough, a slight change is acceptable with their routine.
I hope this message helps.