Black Mare & Sorrel Overo Stallion?

Okay, I'm horrible with color genetics.
What colors are possible for a foal from a black mare (both parents are black) bred to a sorrel overo stallion (both parents are sorrel overo)? Or do you need grandparent's colors from each horse to know?
Also...can someone also explain how you would know if a horse carried the agouti gene or if they were heterozygous/homozygous? Thanks

My guess, if dam was indeed true black on both sides and sire was sorrel overo would be black, possibly with overo.
Black (correct me if I am wrong everyone) is dominant over red.
As for agouti, the only way to know is by getting the horses tested or knowing for certain than neither parent is bay or brown, which in some circumstances can be tricky if the horse is brown but super darkened to the point where it is hard to tell.

Oh, and for the horse to me homozygous for anything he/she would have to have it coming from both sides, not just one

Yes the mare is a true black and her sire & dam as well. I would be interested to know if anyone can verify if black is a dominant gene over red. Thanks for your answer!

Yes black is dominant over red.

What breed is the mare?

What colour are the sire's grandparents?

Subbing because I'd like to hear what the color experts say. I'm pretty sure the only way to know for sure on the agouti and/or homozygous is to have them tested. Please quickly correct me if I'm wrong.

The mare has to have no agouti to be black, so we can definitely rule it out on her side. The stallion, being red based, can "hide" agouti, since it doesn't do anything to a red based horse.

The mare is Arabian.

The stallion's grandpartents are sorrel overo and solid sorrel on both the dam and sire's sides. I attached his pedigree in case what I said is confusing.

Hez Red Hot Paint

You would have to test him to check his agouti status. Unfortunately, it can hide on red horses, and since he goes back to horses with agouti in about the 5th gen, there is no way to tell if that agouti is still there.

Has he sired any other foals?

Yes he has sired other foals, but unfortunately I don't know what the mares' colors were. I know he has foaled mostly sorrel overos, however he did foal one solid chestnut.