Someone told me that we can do Hajj or Umrah without a Mahram if you are in a group of women, as women used to do this in the time of the Prophet (Peace and blessings be upon him) when women do Jihad?

You can find this mentioned in some books and some scholars said this but it is not correct. The Prophet Muhammad (Peace and blessings be upon him), made it very clear, he said, “No woman who believes in Allah and the Last Day, should travel without a Mahram.”

In another Hadith, one of the Sahabah said to the Prophet (Peace and blessings be upon him), “O Prophet of Allah, my wife set off for Hajj (with group of people) and they have registered my name to go for Jihad.” The Prophet Muhammad (Peace and blessings be upon him) said, “Leave the Jihad and join your wife.”

So in Islam it is totally Haram for a woman to travel without a Mahram, even if she does not have a Mahram, the Hajj is canceled; she is exempted from doing the Hajj.