Automorphisms of one-relator groups

It is a well known fact that every group G has a presentation of the form G = F/R, where F is a free group and R the kernel of the natural epimorphism from F onto G. Driven by the desire to obtain a similar presentation of th e group of automorphisms Aut (G), we consider the subgroup Stab (R) subset of or equal to Aut(F) of those automorphisms of F that stabilize R and ask whether the natural homomorphism Stab (R) --> Aut(G) is onto; if it is, we can try to determine its kernel. Both parts of this task are usually quite hard. The former part received co nsiderable attention in the part, whereas the more difficult part (determin ing the kernel) seemed unapproachable. Here we approach this problem for a class of one-relator groups with a special kind of small cancellation condi tion. Then, we address a somewhat easier case of 2-generator (not necessari ly one-relator) groups and determine the kernel of the above-mentioned homo morphism for a rather general class of those groups.