  In this article, we study the elements with disconnected centralizer in the
Brauer complex associated to a simple algebraic group G defined over a finite
field with corresponding Frobenius map F and derive the number of F-stable
semisimple classes of G with disconnected centralizer when the order of the
fundamental group has prime order. We also discuss extendibility of semisimple
characters to their inertia group in the full automorphism group. As a
consequence, we prove that "twisted" and "untwisted" simple groups of type E_6
are "good" in defining characteristic, which is a contribution to the general
program initialized by Isaacs, Malle and Navarro to prove the McKay Conjecture
in representation theory of finite groups.
