  We compute a twisted first cohomology group of the automorphism group of a
free group with coefficients in the abelianization $V$ of the IA-automorphism
group of a free group. In particular, we show that it is generated by two
crossed homomorphisms constructed with the Magnus representation and the Magnus
expansion due to Morita and Kawazumi respectively. As a corollary, we see that
the first Johnson homomorphism does not extend to the automorphism group of a
free group as a crossed homomorphism for the rank of the free group is greater
than 4.
