OK, here is the official policy:
If a baptized man is present, he must should the prayer.
If an unbaptized man is present, then the woman should pray, but she must have a head covering, on account of the angels.
If a baptized man is present, but for some reason cannot say the prayer (perhaps his head was stuck up his ass or some such thing), then the woman can pray, but she must have a head covering.
If no men were present, the women could pray all they wanted with nothing on their heads.
The head covering can be anything. A hat will work, but if there are no hats, then anything at all will suffice - handkerchief, ash tray, coaster, etc.
For some reason, everyone always seemed to use a napkin. I personally saw my mother use one. Did no one have a hat back then? Or, were women making a statement by using something preposterous?
Now, I'm not sure why the presence of a penis made such a difference to the angels. Even a baby boy would be cause for scurrying for a hanky.