The question I want to ask is actually slightly broader than that in the title: what is the smallest class of finite groups which contains all finite simple groups groups, and is closed under semidirect products?
This arose out of a conversation I was having with a friend a while back. At the time, I thought I saw a simple argument that this class of groups was in fact the class of all finite groups, but that argument turned out to be gibberish. (EDIT: As Arturo points out below, this class cannot possibly consist of all finite groups.)
Thanks in advance for the help!