I have nausea when I take the injection for the first day or two but I haven't been physically sick and that's after being on them for 3 months. Every ones body will react differently so I understand your concern. But I agree with @jpscloud the benefits do outweigh the side effects.
I'm now on week 6 of the injections and it does get better. The first 4 weeks I suffered with nausea, upset stomach and heartburn. The side effect of it being an appetite suppressant works, although now I've completely gone off my food and have no fancies... so maybe it's not a positive haha :)