It is not permissible to use equipment belonging to one's work, such as cars and the like, for personal purposes except with permission from the company owner or from one who has the authority to give permission for that.
If your direct supervisor is giving you free rein in using these cars and he has the authority to give them to the employees outside working hours, and he has given you permission to do that on condition that you bear the expenses of fuel and so on, then there is nothing wrong with it, because it is the property of the company and it has the right to dispose of it by means of rental, giving and so on.
This applies if you are working in a private company as you said. But in the case of government institutions, it is not permissible to use their equipment for personal purposes, even if the boss gives permission for that, because he does not have the authority to allow that for himself, so how can he give permission to others to use it? This has been discussed previously in the answer to question number 47067.
And Allah knows best.