It is true that anything supposed to be "Greek" and identified as such in the Greek world,in Sparta it was exagerrated and augmented!Thus,the warriors way,the code of honor and various ideals which are thought to constitute the Greek Way were even more distinct and important in the Spartan case!
I think the same goes for the women,at least where a particular matter comes into focus.To elaborate,it is true that the role of the housewife in Ancient Greece was rather significant due to the fact that men engaged in other activities as exersices and politics.This applied to the case of Sparta as well,but as it was a law that the men should involve themselves only with martial arts,its understood that analogically the role of their wifes became even more significant related as it was to the wellbeing of their estate.Now,if we take the whole female/housewives population and their effect on the maintainancr=e of the State as a whole we can see what enormous their contribution was to the goverment of Sparta.
This was no doubt acknowledged by the males,as they knew that they could not engage carelessly in their exersices and their strategies,if their mind was distracted by affairs of house economy!Thus women aquired an honoured role that was morally invested!The women-wifes that supervised the slave economy(the Spartans could not have been able to surveil all of them-the women must have played a part,without which the slave would have playd havoc!!)became an engine of force for the state,the fuel that fed men's ambitions and allowed them to go on without looking back!They were the stick that kindled the fire of their laws and ideals,warning them against ignominous fault!
Therefore their word must have been heard and respected-that was their right well earned-a kind of informal,unofficial power against matters of the state!This was probably the factual truth Plato had heard of and hich he delivered exagerated(or his source had already blown it of proportion)saying that Spartans are ruled by their wives.
We have to note here that these unofficial privileges relate only to the married and elder women who have been around and are active so as to know and judge the well being of the State!Not to the female sex in general!Reverence for the Elder is applied among women due to the wisdom of ages.Being partly of Laconic descent myself my father used to tell me how powerful a woman his Grandmother from Mani was!She raised a whole family on her own and therefore was unchallenged!When her oldest son who was,at the time,the sole care-taker of the family and a mature man dared to talk back to her,she flung a mag to his head not thinking twice,as his act was interpreted as insolence and disobedience!!The people of Mani still lead a very "Laconic",rigid life,but their women still hold a great position in the family!
To become such a woman of power no doubt the Spartans believed that even as a young girl she had to be nurtured in the Spartan beliefs and ways,so that when time came she would be able to uphold them for the sake of her sons and husbund,and mostly of the state!This idea must have forged their training like men in connection to the eugonical purpose!
Women were admired even more than men when they displayed such manly qualities,as the Greeks were aware that such an attitude was in contradiction to the feeble female nature,and to overcome it was an achievement to be admired even more than male courage,as the latter was more of an obligation expected of a true man.
In conclusion,it is my belief that women had as great a share in forming heroic Sparta as the men.Without their courage and stamina the men would have always looked back in worry-a lethal concern in the heat of battle!