θ-semisimple twisted conjugacy classes of type D in PSL(n,q)

Let p be an odd prime, m is an element of N and set q = p(m), G = PSLn(q). Let theta be a standard graph automorphism of G, d be a diagonal automorphism and Fr-q be the Frobenius endomorphism of PSLn((F-q) over bar). We show that every (d o theta)-conjugacy class of a (d o theta, p)-regular element in G is represented in some Fr-q-stable maximal torus of PSLn((F-q) over bar) and that most of them are of type D. We write out the possible exceptions and show that, in particular, if n >= 5 is either odd or a multiple of 4 and q > 7, then all such classes are of type D. We develop general arguments to deal with twisted classes in finite groups.