Using the definite article "the" with Company Name + Group denominations
You should use the same formulation as the company itself uses. Have a quick look at the "About Us" tab on their websites - for example, ExxonMobil refers to itself as ExxonMobil - no "the" or "Group" - and it's the same with General Motors.
If "Group" is not an official part of the company's name but you want to reflect that the company has various incorporated entities related to it, then it would be appropriate to add "the" before the company name and "group" (not capitalised) after: hence "the ExxonMobil group".