All (well sorta) natural.

There are fish that have dyes injected in them-some tetras and the "glass" catfish. You can tell an injected fish because its color is very extreme, ie neon.
Wild bettas have a great amount of color, but not as much as what you see in the pet store. Pet bettas have been bread selectively over the years to look the way they do. Kinda like getting the poodle from a wolf, except on a smaller scale. Bettas are one of the easier fish to breed, so many people have developed different varieties, which are even more gorgeous than the pet store fish. Do a google on breeding bettas to see.

The short answer: no.

No - bettas are not injected with color. The colors you see are natural, in that that over many years the fish have been selectively bred for them.

they are not injected. I dont know about Betta's, but several types of fish are bred in colored water to make them the color they are.

Those are Betas natural colors. I cant believe that your mother think they are dyed!!! No, they are not dyed!!! Tell your mother to go to any stores that sell Betas and she will see that some Betas does not look pretty.

Some other fish are injected with color [glass fish, ompak cats] but Betta's are selectively bred for their beautiful form and colors.