  We generalize the notion of a Magnus expansion of a free group in order to
extend each of the Johnson homomorphisms defined on a decreasing filtration of
the Torelli group for a surface with one boundary component to the whole of the
automorphism group of a free group $\operatorname{Aut}(F_{n})$. The extended
ones are {\it not} homomorphisms, but satisfy an infinite sequence of
coboundary relations, so that we call them {\it the Johnson maps}. In this
paper we confine ourselves to studying the first and the second relations,
which have cohomological consequences about the group
$\operatorname{Aut}(F_{n})$ and the mapping class groups for surfaces. The
first one means that the first Johnson map is a twisted 1-cocycle of the group
$\operatorname{Aut}(F_{n})$. Its cohomology class coincides with ``the unique
elementary particle" of all the Morita-Mumford classes on the mapping class
group for a surface [Ka1] [KM1]. The second one restricted to the mapping class
group is equal to a fundamental relation among twisted Morita-Mumford classes
proposed by Garoufalidis and Nakamura [GN] and established by Morita and the
author [KM2]. This means we give a simple and coherent proof of the fundamental
relation. The first Johnson map gives the abelianization of the induced
automorphism group $IA_n$ of a free group in an explicit way.
