What were June's parents? Palomino is red plus creme. You really can't tell if a red horse carries Agouti, because Agouti acts by restricting black, and there isn't any black for it to act on in a red-based horse. Black is non-agouti, so if the Friesian is black, you know he doesn't carry agouti - the only possible source for Agouti in this hypothetical cross would be June, and you can't tell by looking whether she has it or not. It's not unheard of for Friesians to carry red, but it's pretty unusual.
You could spend a little money, send some hair off to a lab that does genetic testing and find out what she has; that would help you narrow down the possibilities. But offhand:
Unless June has Agouti, you can rule out Agouti colors like Bay and Buckskin
Unless the Friesian carries red, you can rule out red-based colors like Palomino
So what you are left with is basic black, and black + creme, which is a color sometimes called smoky black
Smoky blacks can vary from not-quite-black to a sort of chocolate color.