For odd primes p, we examine H ̂ ∗(Aut(F 2(p−1)); Z (p)) , the Farrell cohomology of the group of automorphisms of a free group F 2( p−1) on 2( P−1) generators, with coefficients in the integers localized at the prime (p)⊂ Z . This extends results by Glover and Mislin (J. Pure Appl. Algebra 150 (2) (2000)), whose calculations yield H ̂ ∗(Aut(F n); Z (p)) for n∈{ p−1, p} and is concurrent with work by Chen (Farrell cohomology of automorphism groups of free groups of finite rank, Ohio State University Ph.D. Dissertation, Columbus, Ohio, 1998) where he calculates H ̂ ∗(Aut(F n); Z (p)) for n∈{ p+1, p+2}. The main tools used are Ken Brown's “normalizer spectral sequence” (Brown, Cohomology of Groups, Springer, Berlin, 1982), a modification of Krstic and Vogtmann's (Comment. Math. Helv. 68 (1993) 216–262) proof of the contractibility of fixed point sets for outer space, and a modification of the Degree Theorem of Hatcher and Vogtmann (J. London Math. Soc. (2) 58 (3)(1998) 633–655).