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ii) g* x) =x~xgx, for all g, x e G and / e Aut G. We see that Aut G is naturally considered as part of a crossed module: that is, a group homomorphism d: M-+P together with an action of P on M satisfying CM1: d(m p )=p-1 d(m)p, CM2: mo {m) = m~lm o m, for all m 0 , meM andp eP.Crossed modules were introduced by J. H. C. Whitehead [16] and among the standard examples are the inclusion M<-+P of a normal subgroup M of P, the zero homomorphism M->P when M is a P-module, and any surjection Af-»P with central kernel. There is also an important topological example: if F-»£-» # is a fibration sequence of pointed spaces, then the induced homomorphism jiiF-> it X E of fundamental groups is naturally a crossed module. Now groups are algebraic models of 1-types: that is, there is a classifying space functor B: (groups)-• (OV-complexes) such that for any group G the space BG satisfies JZ X BG = G and JtjBG = 0 for j > 1, and further any pointed, connected CW-complex X with jtjX = 0 for j > 1 is of the homotopy type of Bn x X.Crossed modules are algebraic models of 2-types. There is a classifying space functor B: (crossed modules)-» (ClV-complexes)

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We introduce conditions on a group action on a tree that are su½cient for the action to extend to the automorphism group. We apply this to two di¨erent classes of one-relator groups: certain Baumslag±Solitar groups and one-relator groups with centre. In each case we derive results about the automorphism group, and deduce that there are relatively few outer automorphisms.