A woman has no chiyuv whatsoever to say kaddish – BUT – she may do so quietly in Shul, in a manner that does not unduly draw attention to herself. When my parents were niftar, I did not say Kaddish, but my friend who lost her parents and has no brothers or other male relatives, asked a shailah and was permitted to do so. She made the minyan every day and said kaddish from the ezras nashim. I do not advocate for this, but I do understand that there are women who have the need to express themselves when they are in aveilus. Since it is not assur, kol hakovod to them. It is a big commitment, and I have great respect for the men who fulfill that commitment faithfully.
My personal hashkafa is that women should first faithfully and properly do those mitzvos for which they DO have an achrayus, before attempting to take on mitzvos for which they have never had any obligation.