Unfortunately rare because of their preference for dryer climates. Usually 10cm tall or so. Although found on the ground, it is not definitively known if they are mycorrhizal or not.
Phallus impudicus - yes, the Latin means what you think it does, head wider than stem. Large, up to 20cm tall.
You thought those two were rare, these three are extremely rare.
Dictyophora duplicata - like Phallus but with a bridal-like veil!
Lysurus cruciatus - the head is ribbed like a withered claw that had a Harry Potter spell performed on it.
L. periphragmoides - pinkish stem, top more of a lattice than a claw.
There should be a specialized book on this group, I would buy it.