Let \(p\) be a prime, \(q\) be a power of \(p\) and let \(G(q)\) be a semisimple group of Lie type in characteristic \(p\). This paper is a continuation of a paper written by the author [Contemp. Math. 153, 1-9 (1993; Zbl 0823.20013)] aimed at studying extending the Steinberg character \(St_G\) of \(G(q)\) to any group \(H\) containing \(G(q)\) as a normal subgroup. Typical results in this direction are the following: Theorem A: Suppose \(G(q)\triangleleft H\). Then the following hold (i) there exists a \(p\)-group \(D\) in \(H\) which is a complement of an \(S_p\)-group of \(G\) in an \(S_p\)-group of \(H\) such that if \(y\in D\) then \(C_{G(q)}(y)\) has a \(p\)-block of defect 0. (ii) If \(y\) is a \(p\)-element in \(H\) such that \(C_{G(q)}(y)\) has a \(p\)-block of defect 0, then \(y\) is conjugate in \(H\) to an element of \(D\). Theorem C: Suppose that \(G(q)\) is the group of rational points of a simple group of Lie type over the algebraic closure of \(F_p\), and \(G(q)\triangleleft H\) with \(C_H(G(q))\subseteq G(q)\). Let \(D\) be defined as in Theorem A. If \(x=x_px_{p'}=x_{p'}x_p\in H\) with \(p\)-part \(x_p\), then (i) \(\chi(x)=0\) if \(x_p\) is not conjugate to an element of \(D\); (ii) \(\chi(x)=\pm|C_{G(q)}(x)|_p\) if \(x_p\in D\), where \(\chi\) is the extended character of \(St\).