The number of automorphisms of a monolithic finite group

We develop a general formula for the order of the group of automorphisms Aut(G) of a monolithic finite group G in terms of information about the complex characters of G and information about how G is embedded as a subgroup of a particular finite general linear group. We show that this formula is applicable to a wide variety of finite p-groups, and we discuss several applications to particular large families of finite p-groups. Finally, we derive a useful consequence of this formula and discuss its application to a classification problem involving finite p-groups.