Two of my four kids "hold it together" at school but totally break down at home. With my oldest, we found that the school he was in ended up not being a good fit even though he seemed "fairly ok" there. Once we switched to a GT school all was well. Now he is doing great at home and school.
My dd5 could behave at school but at home she was horrible. We tried every behavorial program we could and finally got outside therapy. Turns out she is depressed and probably has been since about age 3. She was always short tempered, irritable sullen, etc. at home. We really lived in total chaos waiting for her next outburst or issue. Life was really very hard to enjoy as a family.
She has been on anti depressants for a month now and I have a completely different child!!! She is calm, caring, more verbal, smiling, etc. It is like I am finally connecting with her for the first time in years and it's the best feeling. We hated to do meds but seeing how much it has helped her (100% difference) I would highly recommend getting some outside help. Not saying meds are needed, just that professionals can really get to the root of this stuff pretty quick.
Good luck
