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Homulilly is Homura Akemi's witch form.
Madoka Magica Portable game
Homulilly is a Witch of Mortal World with a disc on top of her and infinite sign. She wears headwear resembling a stereotypical witch hat. Homulilly first appeared in the bonus material of Madoka Magica Portable. Unlike all other witches in the game, with the exception of Kriemhild Gretchen, Homulilly is not playable in the game or bonus dungeons, nor is any other information provided in the game or bonus materials. She only appears in a Game Over screen in the game if a player loses to Walpurgis Night in Homura's route, under specific conditions. There is a monologue illustrating Homura losing hope to defeat Walpurgis Night and running out of magic to turn back time, and an illustration of Homulilly appears before the Game Over screen.
Her Witch's Kiss from Special PlayStation Portable theme bundled with limited edition of Madoka Magica Portable looks like an hourglass on top of a spider lily, although it is Homura's character color purple, instead of the flower's usual red color.
Puella Magi Madoka Magica Movie: Rebellion Story
WARNING: THE FOLLOWING SECTION CONTAINS HEAVY SPOILERS FOR THE PUELLA MAGI MADOKA MAGICA MOVIE. READ AT YOUR OWN RISK!
The Nutcracker Witch. Its nature is self-sufficiency. Its gallant form, which once split many nuts, is now useless. Without any other purpose, this witch's last wish is her own execution. However, a mere decapitation will not clear away the witch's sins. This foolish witch will forever remain in this realm, repeating the procession to her execution.
The Nutcracker Witch. Her nature is self-sufficiency. Her teeth are showing, her skull is melted, and her eyeballs have fallen out. A promise is the only thing that pitifully planted in that head which can no longer crack any nuts, but within the husk of the awakened witch is the distinct form of a magical girl. Her servants shamefully refer to that thing as a good-for-nothing.
Homulilly, the Nutcracker Witch, is still Homura's Witch form. She was formed as part of an experiment conducted by Kyubey and the Incubators. After Madoka Kaname made her wish to erase Witches from existance, Kyubey became interested in restoring the Witch System, which he believed to be more effective for the production of energy than slaying the Wraiths that replaced the Witches. In order to observe and possibly subvert and control Madoka, the Incubators devised an Inhibition Field wherein they placed a dying Homura. Normally, in such a situation Madoka would have spirited away the girl and erased her despair, but the Inhibition Field made such an intervention impossible. Homura began her transformation into a Witch, but the Field kept it under control. Kyubey hoped that Madoka would personally come to the rescue of the suffering girl, allowing the Incubators to observe her and learn how to deal with her.
Inside her Soul Gem, Homura created a Barrier that was her idealized version of Mitakihara City. This barrier accidentally absorbed Mami Tomoe and Kyouko Sakura, while Madoka and her subordinates Sayaka Miki and Nagisa Momoe entered it willingly. Homura was initially unaware of her own status as a Witch. Upon learning of and Kyubey's plan, she turned into Homulilly (though without actually shattering her Soul Gem and transforming it into a Grief Seed). Her main goal as a Witch was to kill herself so that Madoka would not have to reveal the secrets of her power to Kyubey. However, the Magical Girls inside the gem managed to purify the Witch, also destroying the Inhibition Field and freeing Homura.