  We introduce a notion of quasimorphism between two arbitrary groups,
generalizing the classical notion of Ulam. We then define and study the
category of homogeneous quasigroups, whose objects are groups and whose
morphisms are equivalence classes of quasimorphisms in our sense. We call the
automorphism group QOut(G) of a group G in this category the quasioutomorphism
group. It acts on the space of real-valued homogeneous quasimorphisms on G
extending the natural action of Out(G). We discuss various classes of examples
of quasioutomorphisms of free groups. We use these examples to show that the
orbit of Hom(F_n, R) under QOut(F_n) spans a dense subspace. This is contrast
to the classical fact that the corresponding Out(F_n)-orbit is closed and of
uncountable codimension. We also show that QOut(Z^n) = GL_n(R).
