Final answer:
Haudenosaunee women played critical roles in their government by choosing the members of the Great Council, removing Great Council leaders, and having the power to overrule council decisions. They did not, however, run council meetings or keep records of laws passed by the council.
Step-by-step explanation:
Haudenosaunee women had significant roles in the governance of their communities, and their participation in government differed from many other societies of their time. Despite the sachems, who were male leaders, being the public face of leadership, Haudenosaunee women wielded considerable power behind the scenes.
The women chose the members of the Great Council, which was a clear demonstration of their political influence. Should the need arise, they could also remove any Great Council leaders who did not fulfill their duties appropriately or to the satisfaction of the community. Furthermore, Haudenosaunee women could overrule council decisions, which implies that they had a critical check on the council's power. Their consent was necessary for decisions to take effect, giving them a de facto veto power over the council's deliberations.
However, contrary to some misconceptions, Haudenosaunee women did not keep a record of all laws passed by the council nor did they run Great Council meetings. These responsibilities were typically carried out by the male leaders. Nonetheless, the influential roles of Haudenosaunee women in their government system showcased a balance and sharing of power between genders, which was notable during that historical period.