During the war of Karbala, were the people who killed Imam Hussain R.A. the Yazidi? Which ummah do they belong to, Sunni, Shia, or Wahabi?

I am really happy that someone asked this question.

Here's the thing,

There were no Sunni, Shia, Wahabi branches of Islam as it is today, in those times, people were known by their native place, like there were Kufis from Kufa, Farsi from Persia etc.

So who killed Imam Hussain (A.S)?

I get this from many of Muslims that Shias are the one who killed Imam Hussain and now they're the one weeping for Him(A.S) and that is so untrue.

 So what happened exactly?

When people came to know about how brutal and tyrant Muawiya and Yazid was, they wanted someone more generous ruler like Imam Hussain to rule upon them and they wrote to Imam to come and fight for them and they'll support Him, now these people didn't wanted Imam Hussain as he was chosen by Allah and he was son of Ali just for the sake of good ruler.

There were basically three types of people in Kufa.

1. The one who really loved Imam and were ready to sacrifice their lives for Imam.
2. The one who wanted Imam as the ruler just because he is more generous than a tyrant ruler like yazid.
3. The one who wanted to support Imam because, they thought when Imam will wage a war against Yazid, they'll get a lot of booty as Yazid was an opulent ruler.

But when Yazid sent Ubaidullah ibn Ziyad as the governor of Kufa who was a cruel ruler and didn't spared life of anyone who supported Imam, the Type 2 and Type 3 people got against Imam.

There were Type 1 people in kufa who sacrificed themselves for Imam and bared a lot of pain on the consequences of supporting Imam, those were true Shia, who didn't care for their lives, and believed the Imamate as a divine order and went through all the oppression carried upon them to support the Imam.

So Type 2 and Type 3 kufis killed Imam Hussain(A.S) on the orders of Yazid.