Nice to see you posting, it's been a while. :)
My Eiva did that on one stage. For a few months, my hubby wants to get rid of her. I took her to the vet. The vet asked me to change the cat little to cheap clumping clay. Put her poo on to her tray open uncover. Take her to smell and she quickly cover up by the clumping clay. Once she covered, I scooped it out. She gets back to her little tray till now. Not missed one outside the tray so far.
Hope it works for her.
It sounds like it could be a dominance problem- she is dominating the males of the house by leaving her poop out uncovered. Two of my female cats were fighting for dominance and did this sometimes- but I now believe it was a health issue maybe too. You need to have at least one litter box per cat and have a separate one for her alone. In case it is medical also I would have bloodwork done and see if something else is going on. I hope you don't consider letting your beloved kitty of many years go- I'm sure you can find a way to work it out. Have you tried cat attract litter?
My female cat has been doing this for many years. My male cats have never done this thank god. No matter what I use on the carpet she will still do it. It's not a problem with her food, shes very happy with the food she eats. She never gets any people food either. When she does actually use her litter box she's never in any pain. I have taken her to the vet more times than I can count & they keep telling me that there is nothing wrong with her. I have consulted more than 1 vet just to make sure.
please be patient and find the cause, don't get rid of a long time family member...I will give you a link with some suggestions. however I will also make some of my own suggestions.
its the general rule of thumb to have ONE litter box per cat PLUS one more..therefore your multicat household should have 4...sounds like lots but I think your female is really intimidated by the boys..!!
also rule out a bowel or bladder problem, sometime kitties will attempt to tell us it hurts 'down there' can be a UTI and she is associating it with the litter box....doesn't matter if pain is located in bowel or bladder cause 'down there' could be either, even if its bladder pain she may still communicate this by pooping indiscriminately...hope you understand what I'm 'trying' to say??
http://healthypets.mercola.com/sites/healthypets/archive/2011/06/16/tracking-down-the-cause-of-your-cats-bad-bathroom-habits.aspx
you can also try to use a product avail at your Vet's or a good pet shop or at amazon called FELIWAY this helps to calm down an anxious kitty, its perfectly natural and harmless to use. you can try to room spray and will need to use at least once a day in each area or use the room diffuser.
http://www.amazon.com/s/ref=nb_sb_ss_i_2_7/185-2493458-1658144?url=search-alias%3Dpets&field-keywords=feliway+spray&sprefix=feliway%2Cpets%2C291
also make sure and get a good enzyme carpet cleaner this HAS to be the kitty product with the enzyme needed to remove the offending scent from her sensitive nose, otherwise if she can smell where she's used she will continue to reuse, thinking its an acceptable place to go...also the males will soon start to follow suit, so its very necessary that you get one of these enzyme products and clean the areas well. there are others beside Natures Miracle, some work better than others and may take some trial and error.....you can find these at any good pet store, but do not buy the grocery store brands. good luck and post again if we can help or give more suggestions...please don't give her up.
http://www.amazon.com/Natures-Miracle-Stain-Remover-32-Ounce/dp/B0002ASLMW/ref=sr_1_6?s=pet-supplies&ie=UTF8&qid=1357683646&sr=1-6&keywords=enzyme+miracle
If this is a change for her, I would suspect there is something going on with her health even if the vet can't find it. Chronic irritation of the bowel can do this. My cat once had to be put on prednisone for a similar problem, and it crops up only when he has eaten something that doesn't go down well (I'm suspecting it's when he gets anything with cream in it).