Does a government automatically become fascist if it supports and funds a genocide?

All combinations are possible

Any type of government can do a genocide…

Fascists almost always do, it’s a square/rectangle thing.

No?

Does a government automatically become fascist

Yes

It should, but no. At one point anti-Semitism wasn’t deeply offending. It only tended to start getting offensive when the offending government started invading half the world, and naturally, invading the assets of many millionaires.